From wikipedia entry on Sheaf Cohomology I have found the intriguing passage: 'The essential point is to fix a topological space X and think of cohomology as a functor from sheaves of abelian groups on X to abelian groups.'

I'd very much appreciate if someone could point me to the point in the mathematical literature where Grothendieck introduces this idea of cohomology as functor of sheaves of abelian groups on X to abelian groups. Thanks!

Perhaps somewhere here? Grothendieck's paper Some aspects of homological algebra (1955-1957); Section 3: Cohomology with coefficients in a sheaf, seems a good place to begin the search it seems to me.