understanding how a cohomlogy functor can be a right derived functor of a left exact functor

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My supervisor has been telling me about how you can understand a cohomology functor as a right derived functor of a left exact functor. We were talking about two specific examples, the cohomology of a topological space with respect to a sheaf, and the cohomology of a group with respect to a representation. Is there any source written in English which might help me understand this better, for someone who isn't a huge expert in homological algebra?