I've read about model categories from an Appendix to one of Lurie's papers.
What are the examples of model categories? What should be my intuition about them?
E.g. I understand the typical examples come from taking homotopy of something — but are all model categories homotopy categories?
Here's a bit of the historical reason why model categories came up. If you have a functor on an abelian category which doesn't quite behave as you would like it to, e.g. is not left exact, there is a simple recipe to derive it: 1) Embed your category into the bigger category of complexes 2) Find a qausi-isomorphic replacement for the object you want to plug into the functor. This was so easy because the starting category was abelian. One important example of a category that is not abelian is the category of commutative rings, and there is one important functor on this category, namely the functor of Kaehler differentials, which is not quite exact. (Remember the conormal sequence from Hartshorne, it's only got a zero on one side). So what to do? Since commutative rings are not abelian, there's no way that you can work with complexes. Quillen's great move was to use simplicial objects instead of complexes. This takes care of step 1. You've just succesfully embedded the category of commutative rings into the category of simplicial commutative rings. Now you have to care of step 2. This is possible if the larger category you just moved into is a model category. The replacement is either called cofibrant or fibrant replacement, depending on your functor. The existence of such a replacement follows immediately from the axioms of a model category, since you can factor the map from the initial object to any object into a cofibration and a trivial fibration. Vice versa with final object for fibrant replacement. So we can also do step 2 now! Coming back to the example of the Kaehler differentials, we can derive them now: 1) View your commutative ring as a trivial simplicial ring. 2) Replace it cofibrantly 3)Apply \Omega levelwise. What you get is a simplicial module. If you apply the Dold-Kan correspondence you get complex, and this is the cotangent complex.
Of course there's much more to Model categories, but deriving functors from non-abelian categories is already quite something.
Besides the comments above, it might be useful to have a list of categories that have natural model structures:
It is important to emphasize that these categories often carry several different model structures giving non-equivalent homotopy categories. Another point is that most of these are in some sense built out of simplicial sets. Grothendieck in letters (to Brown?) suggested the idea of a test category, categories which could play the same role as the category Delta. In addition these may come with extra structure. I believe this is the case for cubical objects and similar constructions.
There are also two common operations to do on such model categories: localization and stabilization. The first produces a new model structure on a fixed model category, while the second produces a new category with a model structure.
The idea of localization is to allow us to make certain types of morphisms into isomorphisms in the homotopy category. For instance, in the category of simplicial sets, we may want to consider p-local equivalence. There is a model category structure on simplicial sets such that X -> Y is an isomorphism in the homotopy category if and only if it induces an isomorphism on p-localized homotopy groups. Here, p-localized usually means p-adic. This was first systematically laid out in the book Homotopy limits, completions, and localizations by Bousfield and Kan. A modern account is in Hirschhorn's book Model categories and their localizations.
As for stabilization, this is the process of making the loop space functor invertible. In all of the examples above there exist small homotopy limits, and we can thus form the loop space of any object X as the homotopy fiber product of two maps sending a point to the base-point of X. In this setting we can apply stabilization. For instance, using this on simplicial sets gives us the category of spectra (or, pre-spectra), and the fibrant objects in this category are the Omega-spectra. This category has the incredibly important feature that fiber and cofiber sequences coincide. See the paper Homotopy theory of Gamma-spaces, spectra, and bisimplicial sets by Bousfield and Friedlander.
The more modern versions of spectra are S-modules and symmetric spectra. For the latter, see the paper by Hovey, Shipley, and Smith. These new versions address the following problem. On the normal category of spectra, there is a natural commutative tensor product (the smash product) on the homotopy category of spectra. But, this does not 'lift' to a commutative tensor structure on spectra. The category of symmetric spectra solves this problem by rigidifying, taking into account the action of the symmetric groups on tensors.
Finally, let me note that you don't get anything for free by using model categories. Specifically, while it is often easy to understand what fibrant objects are, the process of fibrant replacement is very very difficult to understand. This is entirely analogous to the point of view of chain complexes: injective objects are easy to understand. But, the injective resolution of some given object is very difficult.
Model categories are 1-categorical presentations of (∞,1)-categories, which you can just think of as categories enriched in topological spaces, such as the category of spaces itself. (Actually, there are conditions on (∞,1)-categories that come from model categories--most importantly they must have all homotopy limits and colimits.) They're particularly convenient for computations, as with any kind of presentation.
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 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.
One can also view model structures as a solution to the problem of when is a localization of a category locally small. In other words when one wants to invert a collection of morphisms the morphisms in the resulting category are equivalence classes of diagrams and there is no guarantee that there will only be a small set of equivalence classes. If the localization can be performed via taking the homotopy category of a model category then one knows that the hom-sets are fine.
Another thing, which is probably more useful in applications, is that a model structure allows one to perform certain constructions such as taking homotopy limits which can't be done just at the level of homotopy categories.
Maybe this isn't what you intended to ask in the last part of your question but it might be helpful. It is not true that every triangulated category comes from a model structure on some category. In the paper Triangulated Categories Without Models (Muro, Schwede, and Strickland) an example of a triangulated category that does not arise via a model structure is given.
It's not really my thing, but my understanding is that the standard model category is CW-complexes with homotopy classes of maps, and that all the definitions (fibration, cofibration, weak equivalence, etc) are trying to abstract the concepts with the same names in this category. Moving a bit more algebraic, I believe that Derived Categories of Abelian Categories are a type of model category, as are categories of simplicial X where X is pretty much anything. What do all of these have in common? We want to try to do homotopy in them, in some sense. So the intuition I've taken away is that something like "homotopy" should work in these categories.
