Localizations or quotients of categories? Motivation: In the classical construction of the derived category of an abelian category, one (roughly) starts with an abelian category $\mathcal{A}$, then considers the quotient category $\mathcal{K}$ of $\mathcal{A}$-chain complexes modulo homotopy and observes this is triangulated; finally, the derived category $\mathcal{D}$ is the localization of the above triangulated category with respect to some morphisms defined in terms of the triangulated structure.
I have always thought as the first process of modding out by homotopy equivalences as being a way to make homotopy-equivalent maps look the same or inverting things which are homotopy equivalent to invertible arrows. Then realized that this is also the aim of localising: force some arrows to become isomorphisms.
Here the question: is there a conceptual frame in which we can see both localisation and modding out by a relation as special cases? Or, are the two concepts better related that my vague interpretation that they both make some arrows invertible? I have seen someone in nLab saying that they are entirely different constructions but that left me a bit disappointed since I still believe a bit in the feeling that at the end of both days I find myself with a new category having the same objects as the old one, and some new isomorphisms. More generally: I realise that if I had a category in my hands and were willing to see some arrows become isomorphisms, I would not know which of the two approaches would be natural in my case, whence the feeling I still have some fog in my mind.
I should add that a colleague of mine observed that he would expect the first construction to be reminiscent of taking closed immersions, and the second to open immersions; which I fully agree to, but would again like to know how to think about these parallels when dealing with a category.
 A: They are completely different constructions which are in general absolutely unrelated but might coincide on some rare occasion. the image of closed immersion vs open immersion is quite good I think, especially if you think about it in terme of localization of a ring: it might happen that localizing a ring is the same as taking a quotient, but this is rather exceptional...
(Edit: this being said, I totally agree with Qiaochu Yuan comments and "absolutely unrelated" was slightly exaggerated ) 
The reason why they give the same things in a lot of example (including chain complexes), giving the idea that they should be related, is because these examples are Quillen model categories and that it is the main result of Quillen's "Homotopical Algebra" (where he defined model categories) that for Quillen model category the localization by weak equialence can be constructed as a quotient of the full subcategory of fibrant-cofibrant objects.
Edit for References: The orginal paper of Quillen (Homotopical algebra) is actually a really good reference on the subject, which give both the general theory and the examples of simplicial sets and of chain complex.
There has been also several book on the subject, I would advise to take a look to Wikipedia which contains already most of the relevant information as well as references to the main books on the subject.
A: The term you're looking for is 'coinverter' (dual of the notion of inverter). This is a weighted colimit in the $Cat$-enriched category $Cat$. See this page on the nLab.
