In essence, one where inference is based on using Bayes theorem to obtain a posterior distribution for a quantity or quantities of interest form some model (such as parameter values) based on some prior distribution for the relevant unknown parameters and the likelihood from the model.

i.e. from a distributional model of some form, f(Xi|θ)
, and a prior p(θ), someone might seek to obtain the posterior p(θ|X)

.

A simple example of a Bayesian model is discussed in this question, and in the comments of this one – Bayesian linear regression, discussed in more detail in Wikipedia here. Searches turn up discussions of a number of Bayesian models here.

But there are other things one might try to do with a Bayesian analysis besides merely fit a model – see, for example, Bayesian decision theory.