What is the difference between fixed effect, random effect and mixed effect models? In simple terms, how would you explain (perhaps with simple examples) the difference between fixed effect, random effect and mixed effect models? 
 A: Not really a formal definition, but I like the following slides: Mixed models and why sociolinguists should use them, from Daniel Ezra Johnson. A brief recap' is offered on slide 4. Although it mostly focused on psycholinguistic studies, it is very useful as a first step.
A: Another very practical perspective on random and fixed effects models comes from econometrics when doing linear regressions on panel data. If you’re estimating the association between an explanatory variable and an outcome variable in a dataset with multiple samples per individual / group, this is the framework you want to use.
A good example of panel data is yearly measurements from a set of individuals of:


*

*$gender_i$ (gender of the $i$th person)

*${\Delta}weight_{it}$ (weight change during year $t$ for person $i$)

*$exercise_{it}$ (average daily exercise during year $t$ for person $i$)


If we’re trying to understand the relationship between exercise and weight change, we’ll set up the following regression:
${\Delta}weight_{it} = \beta_0$$exercise_{it} + \beta_1gender_i + \alpha_i + \epsilon_{it}$


*

*$\beta_0$ is the quantity of interest

*$\beta_1$ is not interesting, we're just controlling for gender with it

*$\alpha_i$ is the per-individual intercept

*$\epsilon_{it}$ is the error term


In a setup like this there is the risk of endogeneity. This can happen when unmeasured variables (such as marital status) are associated with both exercise and weight change. As explained on p.16 in this Princeton lecture, a random effects (AKA mixed effects) model is more efficient than a fixed effects model. However, it will incorrectly attribute some of the effect of the unmeasured variable on weight change to exercise, producing an incorrect $\beta_0$ and potentially a higher statistical significance than is valid. In this case the random effects model is not a consistent estimator of $\beta_0$.
A fixed effects model (in its most basic form) controls for any unmeasured variables that are constant over time but vary between individuals by explicitly including a separate intercept term for each individual ($\alpha_i$) in the regression equation. In our example, it will automatically control for confounding effects from gender, as well as any unmeasured confounders (marital status, socioeconomic status, educational attainment, etc…). In fact, gender cannot be included in the regression and $\beta_1$ cannot be estimated by a fixed effects model, since $gender_i$ is collinear with the $\alpha_i$'s.
So, the key question is to determine which model is appropriate. The answer is the Hausman Test. To use it we perform both the fixed and random effects regression, and then apply the Hausman Test to see if their coefficient estimates diverge significantly. If they diverge, endogeneity is at play and a fixed effects model is the best choice. Otherwise, we’ll go with random effects.
A: I have written about this in a book chapter on mixed models (chapter 13 in Fox, Negrete-Yankelevich, and Sosa 2014); the relevant pages (pp. 311-315) are available on Google Books.  I think the question reduces to "what are the definitions of fixed and random effects?" (a "mixed model" is just a model that contains both). My discussion says a bit less about their formal definition (for which I would defer to the Gelman paper linked by @JohnSalvatier's answer above) and more about their practical properties and utility. Here are some excerpts:

The traditional view of random effects is as a way to do correct statistical tests when some observations are correlated.
We can also think of random effects as a way to combine information from different
  levels within a grouping variable.
Random effects are especially useful when we have (1) lots of levels (e.g., many species
  or blocks), (2) relatively little data on each level (although we need multiple samples from
  most of the levels), and (3) uneven sampling across levels (box 13.1).
Frequentists and Bayesians define random effects somewhat differently, which affects
  the way they use them. Frequentists define random effects as categorical variables whose
  levels are chosen at random from a larger population, e.g., species chosen at random from a
  list of endemic species. Bayesians define random effects as sets of variables whose parameters are [all] drawn from [the same] distribution. The frequentist definition is philosophically coherent,
  and you will encounter researchers (including reviewers and supervisors) who insist on it,
  but it can be practically problematic. For example, it implies that you can’t use species as
  random effect when you have observed all of the species at your field site—since the list
  of species is not a sample from a larger population—or use year as a random effect, since
  researchers rarely run an experiment in randomly sampled years—they usually use either
  a series of consecutive years, or the haphazard set of years when they could get into the
  field. 
Random effects can also be described as predictor variables where you are interested in
  making inferences about the distribution of values (i.e., the variance among the values of
  the response at different levels) rather than in testing the differences of values between
  particular levels. 
People sometimes say that random effects are “factors that you aren’t interested in.”
  This is not always true. While it is often the case in ecological experiments (where variation among sites is usually just a nuisance), it is sometimes of great interest, for example in
  evolutionary studies where the variation among genotypes is the raw material for natural
  selection, or in demographic studies where among-year variation lowers long-term growth
  rates. In some cases fixed effects are also used to control for uninteresting variation, e.g.,
  using mass as a covariate to control for effects of body size.
You will also hear that “you can’t say anything about the (predicted) value of a conditional mode.” This is not true either—you can’t formally test a null hypothesis that the
  value is equal to zero, or that the values of two different levels are equal, but it is still
  perfectly sensible to look at the predicted value, and even to compute a standard error of
  the predicted value (e.g., see the error bars around the conditional modes in figure 13.1).
The Bayesian framework has a simpler definition of random effects. Under a Bayesian approach, a fixed effect is one where we estimate each parameter (e.g., the mean
  for each species within a genus) independently (with independently specified priors),
  while for a random effect the parameters for each level are modeled as being drawn
  from a distribution (usually Normal); in standard statistical notation, $\textrm{species_mean} \sim {\cal N}(\textrm{genus_mean}, \sigma^2_{\textrm{species}})$.
I said above that random effects are most useful when the grouping variable has many
  measured levels. Conversely, random effects are generally ineffective when the grouping
  variable has too few levels. You usually can’t use random effects when the grouping variable has fewer than five levels, and random effects variance estimates are unstable with
  fewer than eight levels, because you are trying to estimate a variance from a very small
  sample. 

A: Fixed effect: Something the experimenter directly manipulates and is often repeatable, e.g., drug administration - one group gets drug, one group gets placebo.
Random effect: Source of random variation / experimental units e.g., individuals drawn (at random) from a population for a clinical trial.
Random effects estimates the variability
Mixed effect: Includes both, the fixed effect in these cases are estimating the population level coefficients, while the random effects can account for individual differences in response to an effect, e.g., each person receives both the drug and placebo on different occasions, the fixed effect estimates the effect of drug, the random effects terms would allow for each person to respond to the drug differently.
General categories of mixed effects - repeated measures, longitudinal, hierarchical, split-plot.
A: There are good books on this such as Gelman and Hill. What follows is essentially a summary of their perspective.
First of all, you should not get too caught up in the terminology. In statistics, jargon should never be used as a substitute for a mathematical understanding of the models themselves. That is especially true for random and mixed effects models. "Mixed" just means the model has both fixed and random effects, so let's focus on the difference between fixed and random.
Random versus Fixed Effects
Let's say you have a model with a categorical predictor, which divides your observations into groups according to the category values.* The model coefficients, or "effects", associated to that predictor can be either fixed or random. The most important practical difference between the two is this:
Random effects are estimated with partial pooling, while fixed effects are not.
Partial pooling means that, if you have few data points in a group, the group's effect estimate will be based partially on the more abundant data from other groups. This can be a nice compromise between estimating an effect by completely pooling all groups, which masks group-level variation, and estimating an effect for all groups completely separately, which could give poor estimates for low-sample groups.
Random effects are simply the extension of the partial pooling technique as a general-purpose statistical model. This enables principled application of the idea to a wide variety of situations, including multiple predictors, mixed continuous and categorical variables, and complex correlation structures. (But with great power comes great responsibility: the complexity of modeling and inference is substantially increased, and can give rise to subtle biases that require considerable sophistication to avoid.)
To motivate the random effects model, ask yourself: why would you partial pool? Probably because you think the little subgroups are part of some bigger group with a common mean effect. The subgroup means can deviate a bit from the big group mean, but not by an arbitrary amount. To formalize that idea, we posit that the deviations follow a distribution, typically Gaussian. That's where the "random" in random effects comes in: we're assuming the deviations of subgroups from a parent follow the distribution of a random variable. Once you have this idea in mind, the mixed-effects model equations follow naturally.
Unfortunately, users of mixed effect models often have false preconceptions about what random effects are and how they differ from fixed effects. People hear "random" and think it means something very special about the system being modeled, like fixed effects have to be used when something is "fixed" while random effects have to be used when something is "randomly sampled". But there's nothing particularly random about assuming that model coefficients come from a distribution; it's just a soft constraint, similar to the $\ell_2$ penalty applied to model coefficients in ridge regression. There are many situations when you might or might not want to use random effects, and they don't necessarily have much to do with the distinction between "fixed" and "random" quantities.
Unfortunately, the concept confusion caused by these terms has led to a profusion of conflicting definitions. Of the five definitions at this link, only #4 is completely correct in the general case, but it's also completely uninformative. You have to read entire papers and books (or failing that, this post) to understand what that definition implies in practical work.
Example
Let's look at a case where random effects modeling might be useful. Suppose you want to estimate average US household income by ZIP code. You have a large dataset containing observations of households' incomes and ZIP codes. Some ZIP codes are well represented in the dataset, but others have only a couple households.
For your initial model you would most likely take the mean income in each ZIP. This will work well when you have lots of data for a ZIP, but the estimates for your poorly sampled ZIPs will suffer from high variance. You can mitigate this by using a shrinkage estimator (aka partial pooling), which will push extreme values towards the mean income across all ZIP codes.
But how much shrinkage/pooling should you do for a particular ZIP? Intuitively, it should depend on the following:


*

*How many observations you have in that ZIP

*How many observations you have overall

*The individual-level mean and variance of household income across all ZIP codes

*The group-level variance in mean household income across all ZIP codes


If you model ZIP code as a random effect, the mean income estimate in all ZIP codes will be subjected to a statistically well-founded shrinkage, taking into account all the factors above. 
The best part is that random and mixed effects models automatically handle (4), the variability estimation, for all random effects in the model. This is harder than it seems at first glance: you could try the variance of the sample mean for each ZIP, but this will be biased high, because some of the variance between estimates for different ZIPs is just sampling variance. In a random effects model, the inference process accounts for sampling variance and shrinks the variance estimate accordingly.
Having accounted for (1)-(4), a random/mixed effects model is able to determine the appropriate shrinkage for low-sample groups. It can also handle much more complicated models with many different predictors. 
Relationship to Hierarchical Bayesian Modeling
If this sounds like hierarchical Bayesian modeling to you, you're right - it is a close relative but not identical. Mixed effects models are hierarchical in that they posit distributions for latent, unobserved parameters, but they are typically not fully Bayesian because the top-level hyperparameters will not be given proper priors. For example, in the above example we would most likely treat the mean income in a given ZIP as a sample from a normal distribution, with unknown mean and sigma to be estimated by the mixed-effects fitting process. However, a (non-Bayesian) mixed effects model will typically not have a prior on the unknown mean and sigma, so it's not fully Bayesian. That said, with a decent-sized data set, the standard mixed effects model and the fully Bayesian variant will often give very similar results.
*While many treatments of this topic focus on a narrow definition of "group", the concept is in fact very flexible: it is just a set of observations that share a common property. A group could be composed of multiple observations of a single person, or multiple people in a school, or multiple schools in a district, or multiple varieties of a single kind of fruit, or multiple kinds of vegetable from the same harvest, or multiple harvests of the same kind of vegetable, etc. Any categorical variable can be used as a grouping variable.
A: I came to this question from here, a possible duplicate.
There are several excellent answers already, but as stated in the accepted answer, there are many different (but related) uses of the term, so it might be valuable to give the perspective as employed in econometrics, which does not yet seem fully addressed here.
Consider a linear panel data model:
$$
y_{it}=X_{it}\delta+\alpha_i+\eta_{it},
$$
the so-called error component model. Here, $\alpha_i$ is what is sometimes called individual-specific heterogeneity, the error component that is constant over time. The other error component $\eta_{it}$ is "idiosyncratic", varying both over units and over time.
A reason to use a random effects approach is that the presence of $\alpha_i$ will lead to an error covariance matrix that is not "spherical" (so not a multiple of the identity matrix), so that a GLS-type approach like random effects will be more efficient than OLS).
If, however, the $\alpha_i$ correlate with the regressors $X_{it}$ - as will be the case in many typical applications - one of the underlying assumptions for consistency of the standard textbook (at least what is standard in econometric textbooks) random effects estimator, viz. $Cov(\alpha_i,X_{it})=0$, is violated. Then, a fixed effect approach which effectively fits such intercepts will be more convincing.
The following figure aims to illustrate this point. The raw correlation between $y$ and $X$ is positive. But, the observations belonging to one unit (color) exhibit a negative relationship - this is what we would like to identify, because this is the reaction of $y_{it}$ to a change in $X_{it}$. 
Also, there is correlation between the $\alpha_i$ and $X_{it}$: If the former are individual-specific intercepts (i.e., expected values for unit $i$ when $X_{it}=0$), we see that the intercept for, e.g., the lightblue panel unit is much smaller than that for the brown unit. At the same time, the lightblue panel unit has much smaller regressor values $X_{it}$.
So, pooled OLS would be the wrong strategy here, because it would result in a positive esimate of $\delta$, as this estimator basically ignores the colors. RE would also be biased, being a weighted version of FE and the between estimator, which regresses the "time"-averages over $t$ onto each other. The latter however also requires lack of correlation of $\alpha_i$ and $X_{it}$.
This bias however vanishes as $T$, the number of time periods per unit (m in the code below), increases, as the weight on FE then tends to one (see e.g. Hsiao, Analysis of Panel Data, Sec. 3.3.2). 

Here is the code that generates the data and which produces a positive RE estimate and a "correct", negative FE estimate. (That said, the RE estimates will also often be negative for other seeds, see above.)
library(Jmisc)
library(plm)
library(RColorBrewer)
# FE illustration
set.seed(324)
m = 8
n = 12

step = 5
alpha = runif(n,seq(0,step*n,by=step),seq(step,step*n+step,by=step))
beta = -1
y = X = matrix(NA,nrow=m,ncol=n)
for (i in 1:n) {
  X[,i] = runif(m,i,i+1)
  X[,i] = rnorm(m,i)
  y[,i] = alpha[i] + X[,i]*beta + rnorm(m,sd=.75)  
}
stackX = as.vector(X)
stackY = as.vector(y)

darkcols <- brewer.pal(12, "Paired")
plot(stackX,stackY,col=rep(darkcols,each=m),pch=19)

unit = rep(1:n,each=m)
# first two columns are for plm to understand the panel structure
paneldata = data.frame(unit,rep(1:m,n),stackY,stackX) 
fe <- plm(stackY~stackX, data = paneldata, model = "within")
re <- plm(stackY~stackX, data = paneldata, model = "random")

The output:
> fe

Model Formula: stackY ~ stackX

Coefficients:
 stackX 
-1.0451 


> re

Model Formula: stackY ~ stackX

Coefficients:
(Intercept)      stackX 
   18.34586     0.77031 

A: Statistician Andrew Gelman says that the terms 'fixed effect' and 'random effect' have variable meanings depending on who uses them. Perhaps you can pick out which one of the 5 definitions applies to your case. In general it may be better to either look for equations which describe the probability model the authors are using (when reading) or write out the full probability model you want to use (when writing).

Here we outline five definitions that we have seen:

*

*Fixed effects are constant across individuals, and random effects vary. For example, in a growth study, a model with random intercepts $a_i$ and fixed slope $b$ corresponds to parallel lines for different individuals $i$, or the model $y_{it} = a_i + b t$. Kreft and De Leeuw (1998) thus distinguish between fixed and random coefficients.


*Effects are fixed if they are interesting in themselves or random if there is interest in the underlying population. Searle, Casella, and McCulloch (1992, Section 1.4) explore this distinction in depth.


*“When a sample exhausts the population, the corresponding variable is fixed; when the sample is a small (i.e., negligible) part of the population the corresponding variable is random.” (Green and Tukey, 1960)


*“If an effect is assumed to be a realized value of a random variable, it is called a random effect.” (LaMotte, 1983)


*Fixed effects are estimated using least squares (or, more generally, maximum likelihood) and random effects are estimated with shrinkage (“linear unbiased prediction” in the terminology of Robinson, 1991). This definition is standard in the multilevel modeling literature (see, for example, Snijders and Bosker, 1999, Section 4.2) and in econometrics.
[Gelman, 2004, Analysis of variance—why it is more important than ever. The Annals of Statistics.]

A: The distinction is only meaningful in the context of non-Bayesian statistics. In Bayesian statistics, all model parameters are "random". 
A: In econometrics, the terms are typically applied in generalized linear models, where the model is of the form 
$$y_{it} = g(x_{it} \beta + \alpha_i + u_{it}). $$ 
Random effects: When $\alpha_i \perp u_{it}$,
Fixed effects: When $\alpha_i \not \perp u_{it}$. 
In linear models, the presence of a random effect does not result in inconsistency of the OLS estimator. However, using a random effects estimator (like feasible generalized least squares) will result in a more efficient estimator. 
In non-linear models, such as probit, tobit, ..., the presence of a random effect will, in general, result in an inconsistent estimator. Using a random effects estimator will then restore consistency. 
For both linear and non-linear models, fixed effects results in a bias. However, in linear models there are transformations that can be used (such as first differences or demeaning), where OLS on the transformed data will result in consistent estimates. For non-linear models, there are a few exceptions where transformations exist, fixed effects logit being one example. 
Example: Random effects probit. Suppose 
$$ y^*_{it} = x_{it} \beta + \alpha_i + u_{it}, \quad \alpha_i \sim \mathcal{N}(0,\sigma_\alpha^2), u_{it} \sim \mathcal{N}(0,1). $$
and the observed outcome is 
$$ y_{it} = \mathbb{1}(y^*_{it} > 0). $$ 
The Pooled maximum likelihood estimator minimizes the sample average of 
$$ \hat{\beta} = \arg \min_\beta N^{-1} \sum_{i=1}^N \log \prod_{t=1}^T  [G(x_{it}\beta)]^{y_{it}} [1 - G(x_{it}\beta)] ^{1-y_{it}}. $$ 
Of course, here the log and the product simplify, but for pedagogical reasons, this makes the equation more comparable to the random effects estimator, which has the form 
$$ \hat{\beta} = \arg \min_\beta N^{-1} \sum_{i=1}^N \log \int \prod_{t=1}^T  [G(x_{it}\beta + \sigma_\alpha a)]^{y_{it}} [1 - G(x_{it}\beta + \sigma_\alpha a )] ^{1-y_{it}} \phi(a) \mathrm{d}a. $$ 
We can for example approximate the integral by randomization by taking $R$ draws of random normals and evaluating the likelihood for each.
$$ \hat{\beta} = \arg \min_\beta N^{-1} \sum_{i=1}^N \log R^{-1} \sum_{r=1}^R \prod_{t=1}^T  [G(x_{it}\beta + \sigma_\alpha a_r)]^{y_{it}} [1 - G(x_{it}\beta + \sigma_\alpha a )] ^{1-y_{it}},\quad a_r \sim \mathcal{N}(0,1). $$ 
The intuition is the following: we don't know what type, $\alpha_i$, each observation is. Instead, we evaluate the product of likelihoods over time for a sequence of draws. The most likely type for observation $i$ will have the highest likelihood in all periods and will therefore dominate the likelihood contribution for that $T$-sequence of observations. 
