What is Prior, posterior, conjugate prior?

In a Bayesian model the parameter $\theta$ is random. Its prior distribution $p(\theta)$ states what is believed before the data; the posterior distribution $p(\theta \mid x) \propto p(x \mid \theta)\, p(\theta)$ is the Bayesian update of it by the likelihood of the data $x$ (Bayes, 1763). A family of priors is a conjugate prior for a likelihood if the posterior stays in the family.