TITLE: Sheaf cohomology: what is it and where can I learn it?
QUESTION [85 upvotes]: As I understand it, sheaf cohomology is now an indispensable tool in algebraic geometry, but was originally developed to solve problems in algebraic topology. I have two questions about the matter.
Question 1. What is sheaf cohomology? I have a vague idea that it has something to do with right derived functors, but this seems rather far removed from the (admittedly very little) cohomology of (co)chain complexes I do know. I would also like to know why sheaf cohomology appears to be so much more fundamental in algebraic geometry than algebraic topology—for instance, I will be taking second courses in algebraic geometry and topology this coming autumn, but sheaf cohomology only appears in the former, suggesting that perhaps sheaf cohomology is not as relevant in basic algebraic topology. (For example, is there an ‘intuitive’ reason why de Rham cohomology cannot be made to work for algebraic varieties?)
Question 2. Are there any good introductions to sheaf cohomology in a general context? I have tried reading Chapter III of Hartshorne, but very little is getting through, perhaps because I'm not yet comfortable with schemes. A different take—perhaps with an emphasis on manifolds, say—may prove more accessible to me, but since I also need to learn it in the context of algebraic geometry, it would be nice if there were a single text which introduces the theory with applications in both subjects.

REPLY [7 votes]: Are there any good introductions to sheaf cohomology in a general context? 

A source is J.-P. Serre's  Faisceaux Algébriques Cohérents. It is very readable. Although the language of algebraic geometry is a bit older, everything carries over to schemes without any problems. A translated version is also available.