Bayesian logit model - intuitive explanation? I must confess that I previously haven't heard of that term in any of my classes, undergrad or grad.
What does it mean for a logistic regression to be Bayesian? I'm looking for an explanation with a transition from regular logistic to Bayesian logistic similar to the following:
This is the equation in the linear regression model: $E(y) = \beta_0 + \beta_1x_1 + ... + \beta_nx_n$.
This is the equation in the logistic regression model: $\ln(\frac{E(y)}{1-E(y)}) = \beta_0 + \beta_1x_1 + ... + \beta_nx_n$. This is done when y is categorical.
What we have done is change $E(y)$ to $\ln(\frac{E(y)}{1-E(y)})$.
So what's done to the logistic regression model in Bayesian logistic regression? I'm guessing it's not something to do with the equation.
This book preview seems to define, but I don't really understand. What is all this prior, likelihood stuff? What is $\alpha$? May someone please explain that part of the book or Bayesian logit model in another way?
Note: This has been asked before but not answered very well I think.
 A: 
What is all this prior, likelihood stuff?

That's what makes it Bayesian. The generative model for the data is the same; the difference is that a Bayesian analysis chooses some prior distribution for parameters of interest, and calculates or approximates a posterior distribution, upon which all inference is based. Bayes rule relates the two: The posterior is proportional to likelihood times prior.
Intuitively, this prior allows an analyst mathematically to express subject matter expertise or preexisting findings. For instance, the text you reference notes that the prior for $\bf\beta$ is a multivariate normal. Perhaps prior studies suggest a certain range of parameters that can be expressed with certain normal parameters. (With flexibility comes responsibility: One should be able to justify their prior to a skeptical audience.) In more elaborate models, one can use domain expertise to tune certain latent parameters. For example, see the liver example referenced in this answer.
Some frequentist models can be related to a Bayesian counterpart with a specific prior, though I'm unsure which corresponds in this case.
A: Logistic regression can be described as a linear combination
$$ \eta = \beta_0 + \beta_1 X_1 + ... + \beta_k X_k $$
that is passed through the link function $g$:
$$ g(E(Y)) = \eta $$
where the link function is a logit function
$$ E(Y|X,\beta) = p = \text{logit}^{-1}( \eta ) $$
where $Y$ take only values in $\{0,1\}$ and inverse logit functions transforms linear combination $\eta$ to this range. This is where classical logistic regression ends.
However if you recall that $E(Y) = P(Y = 1)$ for variables that take only values in $\{0,1\}$, than $E(Y | X,\beta)$ can be considered as $P(Y = 1 | X,\beta)$. In this case, the logit function output could be thought as conditional probability of "success", i.e. $P(Y=1|X,\beta)$. Bernoulli distribution is a distribution that describes probability of observing binary outcome, with some $p$ parameter, so we can describe $Y$ as
$$ y_i \sim \text{Bernoulli}(p) $$
So with logistic regression we look for some parameters $\beta$ that togeder with independent variables $X$ form a linear combination $\eta$. In classical regression $E(Y|X,\beta) = \eta$ (we assume link function to be identity function), however to model $Y$ that takes values in $\{0,1\}$ we need to transform $\eta$ so to fit in $[0,1]$ range.
Now, to estimate logistic regression in Bayesian way you pick up some priors for $\beta_i$ parameters as with linear regression (see Kruschke et al, 2012), then use logit function to transform the linear combination $\eta$, so to use its output as a $p$ parameter of Bernoulli distribution that describes your $Y$ variable. So, yes, you actually use the equation and the logit link function the same way as in frequentionist case, and the rest works (e.g. choosing priors) like with estimating linear regression the Bayesian way.
The simple approach for choosing priors is to choose Normal distributions (but you can also use other distributions, e.g. $t$- or Laplace distribution for more robust model) for $\beta_i$'s with parameters $\mu_i$ and $\sigma_i^2$ that are preset or taken from hierarchical priors. Now, having the model definition you can use software such as JAGS to perform Markov Chain Monte Carlo simulation for you to estimate the model. Below I post JAGS code for simple logistic model (check here for more examples).
model {
   # setting up priors
   a ~ dnorm(0, .0001)
   b ~ dnorm(0, .0001)

   for (i in 1:N) {
      # passing the linear combination through logit function
      logit(p[i]) <- a + b * x[i]

      # likelihood function
      y[i] ~ dbern(p[i])
   }
}

As you can see, the code directly translates to model definition. What the software does is it draws some values from Normal priors for a and b, then it uses those values to estimate p and finally, uses likelihood function to assess how likely is your data given those parameters (this is when you use Bayes theorem, see here for more detailed description).
The basic logistic regression model can be extended to model the dependency between the predictors using a hierarchical model (including hyperpriors). In this case you can draw $\beta_i$'s from Multivariate Normal distribution that enables us to include information about covariance $\boldsymbol{\Sigma}$ between independent variables
$$ \begin{pmatrix} \beta_0  \\ \beta_1 \\ \vdots \\ \beta_k  \end{pmatrix} \sim \mathrm{MVN} \left(
\begin{bmatrix} \mu_0 \\ \mu_1 \\ \vdots \\ \mu_k \end{bmatrix},
\begin{bmatrix} \sigma^2_0 & \sigma_{0,1} & \ldots & \sigma_{0,k} \\
               \sigma_{1,0} & \sigma^2_1 & \ldots &\sigma_{1,k} \\
               \vdots & \vdots & \ddots & \vdots \\
               \sigma_{k,0} & \sigma_{k,1} & \ldots & \sigma^2_k
\end{bmatrix}
\right)$$
...but this is going into details, so let's stop right here.
The "Bayesian" part in here is choosing priors, using Bayes theorem and defining model in probabilistic terms. See here for definition of "Bayesian model" and here for some general intuition on Bayesian approach. What you can also notice is that defining models is pretty straightforward and flexible with this approach.

Kruschke, J. K., Aguinis, H., & Joo, H. (2012). The time has come: Bayesian methods for data analysis in the organizational sciences. Organizational Research Methods, 15(4), 722-752.
Gelman, A., Jakulin, A., Pittau, G.M., and Su, Y.-S. (2008). A weakly informative default prior distribution for logistic and other regression models. The Annals of Applied Statistics, 2(4), 1360–1383.
