What is the difference between "likelihood" and "probability"? The wikipedia page claims that likelihood and probability are distinct concepts.

In non-technical parlance, "likelihood" is usually a synonym for "probability," but in statistical usage there is a clear distinction in perspective: the number that is the probability of some observed outcomes given a set of parameter values is regarded as the likelihood of the set of parameter values given the observed outcomes. 

Can someone give a more down-to-earth description of what this means?  In addition, some examples of how "probability" and "likelihood" disagree would be nice.
 A: Given all the fine technical answers above, let me take it back to language: Probability quantifies anticipation (of outcome), likelihood quantifies trust (in model).
Suppose somebody challenges us to a 'profitable gambling game'. Then, probabilities will serve us to compute things like the expected profile of your gains and loses (mean, mode, median, variance, information ratio, value at risk, gamblers ruin, and so on). In contrast, likelihood will serve us to quantify whether we trust those probabilities in the first place; or whether we 'smell a rat'.

Incidentally -- since somebody above mentioned the religions of statistics -- I believe likelihood ratio to be an integral part of the Bayesian world as well as of the frequentist one:  In the Bayesian world, Bayes formula just combines prior with likelihood to produce posterior.
A: I will give you the perspective from the view of Likelihood Theory which originated with Fisher -- and is the basis for the statistical definition in the cited Wikipedia article.  
Suppose you have random variates $X$ which arise from a parameterized distribution $F(X; \theta)$, where $\theta$ is the parameter characterizing $F$.  Then the probability of $X = x$ would be: $P(X = x) = F(x; \theta)$, with known $\theta$.  
More often, you have data $X$ and $\theta$ is unknown.  Given the assumed model $F$, the likelihood is defined as the probability of observed data as a function of $\theta$: $L(\theta) = P(\theta; X = x)$.  Note that $X$ is known, but $\theta$ is unknown; in fact the motivation for defining the likelihood is to determine the parameter of the distribution.
Although it seems like we have simply re-written the probability function, a key consequence of this is that the likelihood function does not obey the laws of probability (for example, it's not bound to the [0, 1] interval).  However, the likelihood function is proportional to the probability of the observed data.  
This concept of likelihood actually leads to a different school of thought, "likelihoodists" (distinct from frequentist and bayesian) and you can google to search for all the various historical debates.  The cornerstone is the Likelihood Principle which essentially says that we can perform inference directly from the likelihood function (neither Bayesians nor frequentists accept this since it is not probability based inference).  These days a lot of what is taught as "frequentist" in schools is actually an amalgam of frequentist and likelihood thinking.  
For deeper insight, a nice start and historical reference is Edwards' Likelihood.  For a modern take, I'd recommend Richard Royall's wonderful monograph, Statistical Evidence: A Likelihood Paradigm.  
A: If I have a fair coin (parameter value) then the probability that it will come up heads is 0.5.  If I flip a coin 100 times and it comes up heads 52 times then it has a high likelihood of being fair (the numeric value of likelihood potentially taking a number of forms).
A: Suppose you have a coin with probability $p$ to land heads and $(1-p)$ to land tails. Let $x=1$ indicate heads and $x=0$ indicate tails. Define $f$ as follows
$$f(x,p)=p^x (1-p)^{1-x}$$
$f(x,2/3)$ is probability of x given $p=2/3$, $f(1,p)$ is likelihood of $p$ given $x=1$. Basically likelihood vs. probability tells you which parameter of density is considered to be the variable

A: As far as I'm concerned, the most important distinction is that likelihood is not a probability (of $\theta$).
In an estimation problem, the X is given and the likelihood 
 $P(X|\theta)$ describes a distribution of X rather than $\theta$. That is, $\int P(X|\theta) d\theta$ is meaningless, since likelihood is not a pdf of $\theta$, though it does characterize $\theta$ to some extent.
A: The answer depends on whether you are dealing with discrete or continuous random variables. So, I will split my answer accordingly. I will assume that you want some technical details and not necessarily an explanation in plain English.
Discrete Random Variables
Suppose that you have a stochastic process that takes discrete values (e.g., outcomes of tossing a coin 10 times, number of customers who arrive at a store in 10 minutes etc). In such cases, we can calculate the probability of observing a particular set of outcomes by making suitable assumptions about the underlying stochastic process (e.g., probability of coin landing heads is $p$ and that coin tosses are independent).
Denote the observed outcomes by $O$ and the set of parameters that describe the stochastic process as $\theta$. Thus, when we speak of probability we want to calculate $P(O|\theta)$. In other words, given specific values for $\theta$, $P(O|\theta)$ is the probability that we would observe the outcomes represented by $O$.
However, when we model a real life stochastic process, we often do not know $\theta$. We simply observe $O$ and the goal then is to arrive at an estimate for $\theta$ that would be a plausible choice given the observed outcomes $O$. We know that given a value of $\theta$ the probability of observing $O$ is $P(O|\theta)$. Thus, a 'natural' estimation process is to choose that value of $\theta$ that would maximize the probability that we would actually observe $O$. In other words, we find the parameter values $\theta$ that maximize the following function:
$L(\theta|O) = P(O|\theta)$
$L(\theta|O)$ is called the likelihood function. Notice that by definition the likelihood function is conditioned on the observed $O$ and that it is a function of the unknown parameters $\theta$.
Continuous Random Variables
In the continuous case the situation is similar with one important difference. We can no longer talk about the probability that we observed $O$ given $\theta$ because in the continuous case $P(O|\theta) = 0$. Without getting into technicalities, the basic idea is as follows:
Denote the probability density function (pdf) associated with the outcomes $O$ as: $f(O|\theta)$. Thus, in the continuous case we estimate $\theta$ given observed outcomes $O$ by maximizing the following function:
$L(\theta|O) = f(O|\theta)$
In this situation, we cannot technically assert that we are finding the parameter value that maximizes the probability that we observe $O$ as we maximize the PDF associated with the observed outcomes $O$. 
A: $P(x|\theta)$ can be seen from two points of view:


*

*As a function of $x$, treating $\theta$ as known/observed. If $\theta$ is not a random variable, then $P(x|\theta)$ is called the (parameterized) probability of $x$ given the model parameters $\theta$, which is sometimes also written as $P(x;\theta)$ or $P_{\theta}(x)$.
If $\theta$ is a random variable, as in Bayesian statistics, then $P(x|\theta)$ is a conditional probability, defined as ${P(x\cap\theta)}/{P(\theta)}$.

*As a function of $\theta$, treating $x$ as observed. For example, when you try to find a certain assignment $\hat\theta$ for $\theta$ that maximizes $P(x|\theta)$, then $P(x|\hat\theta)$ is called the maximum likelihood of $\theta$ given the data $x$, sometimes written as $\mathcal L(\hat\theta|x)$. So, the term likelihood is just shorthand to refer to the probability $P(x|\theta)$ for some data $x$ that results from assigning different values to $\theta$ (e.g. as one traverses the search space of $\theta$ for a good solution). So, it is often used as an objective function, but also as a performance measure to compare two models as in Bayesian model comparison. 


Often, this expression is still a function of both its arguments, so it is rather a matter of emphasis.
A: This is the kind of question that just about everybody is going to answer and I would expect all the answers to be good.  But you're a mathematician, Douglas, so let me offer a mathematical reply.
A statistical model has to connect two distinct conceptual entities: data, which are elements $x$ of some set (such as a vector space), and a possible quantitative model of the data behavior.  Models are usually represented by points $\theta$ on a finite dimensional manifold, a manifold with boundary, or a function space (the latter is termed a "non-parametric" problem).
The data $x$ are connected to the possible models $\theta$ by means of a function $\Lambda(x, \theta)$.  For any given $\theta$, $\Lambda(x, \theta)$ is intended to be the probability (or probability density) of $x$.  For any given $x$, on the other hand, $\Lambda(x, \theta)$ can be viewed as a function of $\theta$ and is usually assumed to have certain nice properties, such as being continuously second differentiable.  The intention to view $\Lambda$ in this way and to invoke these assumptions is announced by calling $\Lambda$ the "likelihood."
It's quite like the distinction between variables and parameters in a differential equation: sometimes we want to study the solution (i.e., we focus on the variables as the argument) and sometimes we want to study how the solution varies with the parameters.  The main distinction is that in statistics we rarely need to study the simultaneous variation of both sets of arguments; there is no statistical object that naturally corresponds to changing both the data $x$ and the model parameters $\theta$.  That's why you hear more about this dichotomy than you would in analogous mathematical settings.
A: I'll try and minimise the mathematics in my explanation as there are some good mathematical explanations already.
As Robin Girard comments, the difference between probability and likelihood is closely related to the difference between probability and statistics. In a sense probability and statistics concern themselves with problems that are opposite or inverse to one another.
Consider a coin toss. (My answer will be similar to Example 1 on Wikipedia.) If we know the coin is fair ($p=0.5$) a typical probability question is: What is the probability of getting two heads in a row. The answer is $P(HH) = P(H)\times P(H) = 0.5\times0.5 = 0.25$.
A typical statistical question is: Is the coin fair? To answer this we need to ask: To what extent does our sample support the our hypothesis that $P(H) = P(T) = 0.5$?
The first point to note is that the direction of the question has reversed. In probability we start with an assumed parameter ($P(head)$) and estimate the probability of a given sample (two heads in a row). In statistics we start with the observation (two heads in a row) and make INFERENCE about our parameter ($p = P(H) = 1- P(T) = 1 - q$).
Example 1 on Wikipedia shows us that the maximum likelihood estimate of $P(H)$ after 2 heads in a row is $p_{MLE} = 1$. But the data in no way rule out the the true parameter value $p(H) = 0.5$ (let's not concern ourselves with the details at the moment). Indeed only very small values of $p(H)$ and particularly $p(H)=0$ can be reasonably eliminated after $n = 2$ (two throws of the coin). After the third throw comes up tails we can now eliminate the possibility that $P(H) = 1.0$ (i.e. it is not a two-headed coin), but most values in between can be reasonably supported by the data. (An exact binomial 95% confidence interval for $p(H)$ is 0.094 to 0.992.
After 100 coin tosses and (say) 70 heads, we now have a reasonable basis for the suspicion that the coin is not in fact fair. An exact 95% CI on $p(H)$ is now 0.600 to 0.787 and the probability of observing a result as extreme as  70 or more heads (or tails) from 100 tosses given $p(H) = 0.5$ is 0.0000785.
Although I have not explicitly used likelihood calculations this example captures the concept of likelihood: Likelihood is a measure of the extent to which a sample provides support for particular values of a parameter in a parametric model.
A: do you know the pilot to the tv series "num3ers" in which the FBI tries to locate the home base of a serial criminal who seems to choose his victims at random?
the FBI's mathematical advisor and brother of the agent in charge solves the problem with a maximum likelihood approach. first, he assumes some "gugelhupf shaped" probability $p(x|\theta)$ that the crimes take place at locations $x$ if the criminal lives at location $\theta$. (the gugelhupf assumption is that the criminal will neither commit a crime in his immediate neighbourhood nor travel extremely far to choose his next random victim.) this model describes the probabilities for different $x$ given a fixed $\theta$. in other words, $p_{\theta}(x)=p(x|\theta)$ is a function of $x$ with a fixed parameter $\theta$.
of course, the FBI doesn't know the criminal's domicile, nor does it want to predict the next crime scene. (they hope to find the criminal first!) it's the other way round, the FBI already knows the crime scenes $x$ and wants to locate the criminal's domicile $\theta$.
so the FBI agent's brilliant brother has to try and find the most likely $\theta$ among all values possible, i.e. the $\theta$ which maximises $p(x|\theta)$ for the actually observed $x$. therefore, he now considers $l_x(\theta)=p(x|\theta)$ as a function of $\theta$ with a fixed parameter $x$. figuratively speaking, he shoves his gugelhupf around on the map until it optimally "fits" the known crime scenes $x$. the FBI then goes knocking on the door in the center $\hat{\theta}$ of the gugelhupf.
to stress this change of perspective, $l_x(\theta)$ is called the likelihood (function) of $\theta$, whereas $p_{\theta}(x)$ was the probability (function) of $x$. both are actually the same function $p(x|\theta)$ but seen from different perspectives and with $x$ and $\theta$ switching their roles as variable and parameter, respectively.
A: If we put the conditional probability interpretation aside, you can think it in this way:


*

*In probability you usually want to find the probability of a possible event  based on a model/parameter/probability distribution, etc.

*In likelihood you have observed some outcome, so you want to find/create/estimate the most likely source/model/parameter/probability distribution from which this event has raised. 
A: Likelihood is bound to the statistical model that you have chosen. Let's take a discrete example, and assume you have a single observation. Hypothetically, you could always choose a statistical model that always produces one outcome, the observation that you have, with probability $1$, hence, the likelihood will also be $1$. This would be a bad model but it would fit your data perfectly. So, likelihood, in essence, is a subjective value because it depends on how you want to model your data.
PS: Above is the case when you have a single observation. Similar example can be provided for the case when you have multiple observations, i.e. data. Again, hypothetically, you can restrict your model in such a way that it produces only from the observations that you have. For example, say your observations are the following coin flips, TTHH, TTHT, TTTH, TTTT, then you can restrict the model so that it always produces TT as the first two flips. This will be your assumption, hence it is subjective, and the likelihood you get will be higher than if you had not imposed that restriction.
