A model category (introduced by Quillen in the sixties) is a category equipped with three distinguished classes of morphisms (weak equivalences, fibrations and cofibrations) satisfying some axioms designed to mimic the structure of the category of topological spaces. They provide a framework to study homotopy theory which applies to topological spaces (algebraic topology), chain complexes (homological algebra), and much more. In particular the notion of Quillen equivalence allows one to say when two categories are the same "up to homotopy".
What are good places (book, lecture notes, article...) to learn about model categories?
Assume that the reader is reasonably familiar with category theory, and perhaps with the basics of algebraic topology or homological algebra if some motivation is needed. I would also be interested in more advanced books that deal with finer points of model category theory (localizations come to mind).
This question is inspired by this previous one where the OP asked whether to learn $\infty$-category theory. I believe knowledge of model categories is rather essential to learn $\infty$-category, as for example many coherence results are stated in the form of "Such and such categories of $\infty$-categories are Quillen equivalent". As far as I understand much of $\infty$-category is also designed with model categories in mind, basically "what happens if I'm in a category of bifibrant objects and derived hom spaces". See also the MO question Do we still need model categories?
If possible try to argue why the book/paper you're mentioning is a good place to learn (and not just assert so), please. By now I think the subject is settled enough that there are several good books about it, and I think it would be nice to condense information about all these books in one place, instead of relying on hearsay.