TITLE: Why do we need model categories?
QUESTION [37 upvotes]: I cannot give a good answer to this question. And
2) Why this definition of model category is the right way to give a philosophy of homotopy theory? Why didn't we use any other definition?
3) Has model category been used substantially in any area not related to algebraic topology?

REPLY [6 votes]: Extensive answers have already been given in this thread. Just a few remarks here and there.

I think the question “why we need” assumes something about “we”, and in some extent, about “need”. There are people who study those objects for their own sake, there are those who study applied PDE and have no need of model categories. Even those who enter the domain which can be covered by model categories often work in a specialised situation where more adapted techniques exist (homological algebra and triangulated categories for algebraic geometry being an example). I guess the meta-reason is that at the current historic moment (or maybe that will remain so indefinitely) any description of entities ‘’up to homotopy’’ from scratch cannot be done without having this ‘’up to homotopy’’ notion already defined. One has to break the vicious circle and pull things down from their ‘’platonic’’ world. Which brings me to the answer for 
Since categorical philosophy has proven itself to be reasonably natural, introducing categories with weak equivalences is both a natural step and something reflected in many examples. As we now know, this principle of modelling homotopy phenomena is, formally speaking, just as good as any other higher-categorical approach. Working with an arbitrary category with weak equivalences $(C,W)$ can be impossible in practice, but there are many ways to (homotopically) embed it into a model category.  One can consider the category of simplicial presheaves on $C$ for instance, with a suitable Bousfield localisation of the projective model structure. (The examples usually present themselves with better, often canonical, embeddings.) One can thus work with the objects of $C$ by performing operations on the bigger model category and verifying that the answer is sensible for $C$. This adds to the explanation as why the notion is quite ubiquitous. 
Algebraic geometry (both the field and the community) has different tradition from algebraic topology, yet model categories have found their way here as well. For example, Kontsevich defines a noncommutative space as a suitable DG-category, and considers them up to Morita equivalence. There are a few model structures on DG-categories, which cover both the usual DG- and Morita equivalences, and they have been used to get various results, such as the theorem representing DG-functors between quasicoherent sheaf categories as bimodules. Another issue is the structure of noncommutative cohomological invariants, Hochschild cohomology, Deligne conjecture, and related matters. Many of the results (including laying a foundation of derived algebraic geometry) were obtained by people outside of the core algebraic topology community, with the use of model categories.