I would like to learn about derived functors. Which reference do you advise ?
I have to agree strongly with Ame's answer, in part. Weibel is a great place to go for the formalism. Once you have a little bit of the formalism, though, where to go depends on interests. To really see a lot of the power of derived functors, Hartshorne chapter 3 has some good theorems and exercises using them (though he does bounce back and forth with Cech cohomology...) and Huybrecht's "Fourier Mukai Transforms in Algebraic Geometry" book is very clear (at least to me) in the first few chapters, where he discusses derived categories (Note: I do disagree with Harry, this is a SECOND step, not a first for most) and makes extensive use of them, as he's interested in talking about derived categories, and they're the most natural thing in the universe there.
Also, the following papers might help:
Fourier Mukai Transforms and Applications to String Theory talks about how FM transforms (which are compositions of derived functors) help in string theory
Derived categories for the working mathematician This is a wonderful paper, very clear about what the derived category is, and might help in conjunction with the books above.
The classic book "An introduction to homological algebra" by Weibel is an excellent reference, although it is not an easy read. I would certainly not recommend the book by Kashiwara and Schapira. Rotman is decent, but not as clear as Weibel, and it is full of typos.
When learning about derived functors, also try to see how they are used in an "applied context" (algebraic geometry/algebraic topology/any other applications which might interest you...).
I don't think going straight to derived categories (as Harry Gindi suggests) is a good idea.
It's perhaps in the "second step" category of references, but the book by Gelfand and Manin is a classic reference for derived categories. (Later editions have fewer typos I think).
Rotman - "Introduction to homological algebra"
Peter Hilton and Urs Stammbach's Introduction to Homological Algebra is a great, more or less slow paced introduction to the classical theory. MacLane's Homology also makes a great introduction, provided you have a little background to appreciate its examples.
IMO it is not a good idea to start with derived categories. In most cases that approach results in the quite wrong idea that homological algebra is an absolutely abstract and obstruse subject with no contact with 'concrete math'.
There are some nice lecture notes by Jon Woolf on derived categories, derived functors and triangulated categories.
It's only three lectures'-worth, and the emphasis is more on derived and triangulated categories than derived functors. Still, maybe you'll find them useful. The lectures were for an audience of category theorists, but Woolf is a mostly-algebraic geometer and they finish with a brief description of some advanced applications to geometry.
Oh, and I like Weibel too.
If you already know a cohomology theory I'd go straight to derived categories: it's what I did and, although it wasn't easy, it was sure worth it. Richard Thomas's notes suggested by Charles Siegel are excellent, as a sort of supplement I'd recommend Behrang Noohi's notes http://www.mth.kcl.ac.uk/~noohi/papers/Tehran.pdf
As far as books go, I definitely suggest reading the last chapter of Aluffi's great book Algebra Chapter 0. It doesn't reach derived categories, but it gives you every tool you need to approach them later.
After those I'd say the first chapter of Kashiwara-Schapira's Sheaves on Manifolds is very good, as it is fast and concise. Gelfand-Manin is also worth checking out, although full of mistakes. But if you're really brave I'd read Kashiwara-Schapira's Categories and Sheaves: it's the best (but only if you need all the gory details).