- Interpretation of cohomology of small degree:
$H^1(G,A)$ = crossed homomorphisms $G\to A$ modulo principal ones.
$H^2(G,A)$ = equivalence classes of extensions of G by A.
$H^3(G,Center(G))$ = obstructions to existence of extensions of G by A.
2. Transfer and its applications: If $G$ is finite then
1) $H^i(G,M)$ is a torsion group annihilated by multiplication by $|G|$.
2) Embedding of $p$-primary component of $H^i(G,M)$ into a subgroup of $H^i(P,M)$, for any $p$-Sylow subgroup $P\subset G$.
3. In general, Brown's book "Cohomology of groups" gives a decent overview of what is good to know.