Motivation for Schur multipliers What are Schur multipliers good for?
I should probably clarify what I want. Here is an instructive story of how I came to appreciate complex representations and characters of groups. Basically, I always felt that this was something that I should learn, a "trendy" "cool" topic. But that was not real interest. Real interest came when I heard about Burnside's pq-theorem. This is a cool fact that doesn't explicitly mention representations, but you need to use representations to prove it. This is enough motivation to really get me interested.
Is there a similar thing for Schur multipliers (or group (co)homology in general)? Is there a classic problem that you cannot solve without using group homology, but not about homology per se? Any suggestions are welcome.
Also, if you know of a book that can answer my question, please tell me.
 A: This is not about groups, but Lie algebras. Please keep in mind that by trade I am a group theorist, which happened to venture in the field of graded (over the positive integers) Lie algebras, discovering that they do not look too different from (residually) nilpotent groups.
Anyway, I was studying certain twisted loop algebras, trying too prove they were finitely presented. Turned out they weren't. And the reason was that the finite-dimensional Lie algebra I was starting from had a non-trivial Schur multiplier. This Schur multiplier kept popping up at each loop iteration, making a finite presentation impossible.
So my moral from this little story is that you study Schur multipliers because you cannot avoid them - they appear while you do other things, so you'd better face them.
To address more directly your question, though, there are results such as the Schur-Zassenhaus theorem, or the result of Gaschütz that a non-abelian, finite $p$-group has an outer automorphism, that benefit from the use of cohomology.
A: You can find many applications of cohomology in the book
K.Brown, Cohomology of groups,
in particular, in exercises (there is Russian translation).
As to Schur multiplier, there is a very comprehensive book
Karpilovsky, Gregory: The Schur Multiplier, Oxford University Press, 1987
(it is in Internet).
