E.g. I understand the typical examples are homotopy categories of something — but are all model categories homotopy categories?
This isn't quite right. A model category is a category from which you can construct a nicely behaved homotopy category, by inverting weak equivalences and then passing to homotopy equivalence classes of maps. So, for example, the category of (reasonable) topological spaces has a nice homotopy category, and the category of complexes of coherent sheaves on a scheme has a derived category. (You might that say that what you really want is the homotopy category, and that the model category is merely a good way of representing the homotopy category. Certainly in Lurie's stuff, the homotopy type is what counts.)
Dwyer & Spalinski's review (http://hopf.math.purdue.edu/Dwyer-Spalinski/theories.pdf) is a reasonable explanation of the basics of the theory of model categories. But I've found that, to get anything out of that theory, I needed to have an example very firmly in mind. One good example is Pete May's Concise Course in Algebraic Topology (http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf), which presents the basics of algebraic topology in manner that reflects the model structure.