Derived Category of Coherent Sheaves on Elliptic Curves I know little about algebraic geometry, however while studying noncommutative geometry some results showed that a category I understand well (holomorphic vector bundles over noncommutative tori) was equivalent to the derived category of coherent sheaves over an elliptic curve. So, what is the category of coherent sheaves on an elliptic curve, what is its derived category, and why are these important for algebraic geometry? Edit: Is the category of coherent sheaves on a higher dimensional abelian variety much more complicated than the category of coherent sheaves on an elliptic curve?
I don't necessarily need the most technical account; I'm really just looking to get a sense of why these things are important and what information they encode.
Edit: Since the original post seemed to imply that I did no prior research, here is what I know. The category of coherent sheaves is an expansion of the category of holomorphic vector bundles on an elliptic curve so that the category becomes abelian. This is what allows you to take the derived category. There is also an intrinsic characterization of coherent sheaves, which is like the characterization of a holomorphic vector bundle as a locally free sheaf of $O_X$ modules but loosens the locally free condition. I wasn't looking for textbook definitions, I was looking for intuition that would help me to understand why we care about this, including applications in classical complex geometry. 
 A: The following survey article by Orlov is perhaps the best introduction to this subject.
D. O. Orlov, Derived categories of coherent sheaves and equivalences between them, Uspekhi Mat. Nauk, 2003, Vol. 58, issue 3(351), pp. 89–172, English translation (PDF)
There are also many other accounts, for example the book by Huybrechts.
D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry, Oxford University Press, USA, 2006.
A: This is more a comment than an answer but I dont have enough points to add a comment. You asked:
"I'm really just looking to get a sense of why these things are important and what information they encode."
Coherent sheaves over scheme are locally just (finitely generated) modules over a ring call it R. People are typically interested in understanding rings, or on the geometric level, schemes, and one can  study them directly, but for some reason it turns out to be very helpful to study them indirectly by studying \emph{modules } over $R$. This I think is Morita theory or even just the whole philosophy behind representation theory. If you are going to study modules, then you often can understand them by taking free resolutions, etc, and do homological algebra, and then from homological algebra you are led fairly directly to the derived category of modules (or their global analogue, (quasi)coherent sheaves). And so we can recover a lot of information about a scheme by studying it noncommutative shadow, the derived category of coherent sheaves on it. And for example, there are various famous results like if two smooth schemes $X, Y$ have isomorphic derived categories of coherent sheaves, then are $X$ and $Y$ isomorphic? Well, no in general, but in various cases i.e. under some hypotheses on dimension or canonical bundle, the answer is yes.
