Model categories are an axiomization of the machinery underlying the study of topological spaces up to homotopy equivalence. They consist of a category $C$, together with three distinguished classes of morphism: Weak Equivalences, Fibrations, and Cofibrations. There are then a series of axioms this structure must satisfy, to guarantee that the classes behave analogously to the topological maps they are named after. The axioms can be found here.
(As far as I know...) The main practical advantage of this machinery is that it gives a rather concrete realization of the localization category $C/\sim$ where the Weak Equivalences have been inverted, which generalizes the homotopy category of topological spaces. The main conceptual advantage is that it is a first step towards formalizing the concept of "a category enriched over topological spaces".
A discussion of examples and intuition can be found at this question.
The examples found in the answers to Ilya's question, as well as in the introductory papers I have read, all have a model category structure that could be expected. They are all examples along the lines of topological spaces, derived categories, or simplicial objects, which are all conceptually rooted in homotopy theory and so their model structures aren't really surprising.
I am hoping for an example or two which would elicit disbelief from someone who just learned the axioms for a model category. Along the lines of someone who just learned what a category being briefly skeptical that any poset defines a category, or that '$n$-cobordisms' defines a category.