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Let $G$ be a discrete group, and $BG$ is the classifying space.
It is well-known that the group cohomology of $G$-module M, is the same as the cohomology on $BG$ with coefficient in $\tilde{M}$, which is the associated sheaf of $M$.

Can someone explain how these two cohomologies are related?

$\begingroup$I think the prevailing view is that things covered by Wikipedia and easily found in textbooks are not fair game for MO. See en.wikipedia.org/wiki/Group_cohomology and its textbook references.$\endgroup$

$\begingroup$This is under "MO is not an encyclopedia" in the FAQ. You'll get better answers if you ask more specific questions (for example, if you try to read one of the references, and get confused about a particular point, you can ask about that particular thing).$\endgroup$