TITLE: What is sheaf cohomology intuitively?
QUESTION [85 upvotes]: What is sheaf cohomology intuitively? 
For local systems it is ordinary cohomology with twisted coefficients. But what 
if the sheaf in question is far from being constant? 
Can one still understand sheaf cohomology in some "geometric" way?
For example I would be very interested in the case of coherent $\mathcal{O}_X$-Modules. 
Or even just line bundles.

REPLY [12 votes]: This sounds like a cop-out, I know... what it is not very far from reality, at least in some contexts:
One is not really interested in the higher cohomology groups but only in the actual group of global sections, and higher cohomology groups' purpose in life is to allow us to do inductive proofs and other tricks like dimension shift.