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I apologize in advance if these questions are broad or basic. I tried to read about them at the Wikipedia, but everything is written in the language of category theory, in which I have had no formal training. I am looking for an answer in group theoretic terms as much as possible, with intuition behind the parts which cannot translate.

First, which results has nonabelian group cohomology produced for finite group theory? Has it been used to solve problems for which standard group theoretic tools didn't work (like Burnside and Frobenius' theorems in representation theory) or is it more like category theory, which (from what I understand) is more about a 'way of looking at things' than big nuclear theorems?

Second, how exactly does one go about calculating group cohomologies? Specifically, does a precise algorithmic method exist? If so, is there a 'human' method different from how computers would do it?

Third, what exactly is the difference between this and group homology? Are they intrinsically related disciplines - i.e., do people who study cohomology pretty much also study homology, and vice versa?

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You could take a look at the chapter "The Theory of Group Extensions" in Derek Robinson's book A Course in the Theory of Groups. One section is called "Group-Theoretic Interpretations of the (Co)homology groups". –  j.p. Jan 19 '13 at 14:15

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