Applications of group cohomology to algebra

Here's a simple example off the top of my head. A group is said to be finitely presentable if it has a presentation with finitely many generators and relations. This, in particular, implies that $H_2(G)$ is of finite rank. (You can take nontrivial coefficient systems here too.) So you get a nice necessary condition for finite presentability.

The proof of this fact is simple. If $G$ is finitely presented, you can build a finite $2$-complex that has $G$ as its fundamental group. To get an Eilenberg-Maclane space $K(G,1)$ you add $3$-cells to kill all $\pi_2$, then you add $4$-cells to kill all $\pi_3$ etc... You end up building a $K(G,1)$ with a finite $2$-skeleton.

In addition to what Grumpy Parsnip said about group homology, here's another application: In the field of pro-$p$-groups we have that group cohomology is an extremely useful tool for determining the structure of a group, e.g. finding the numbers of generators and relations of a pro-$p$-group:

Then the generator rank $d(G) = \dim_{\mathbb{F}_p} H^1(G,\mathbb{F}_p)$ and the relation rank $r(G) = \dim_{\mathbb{F}_p} H^2(F, \mathbb{F}_p)$ for a pro-$p$-group $G$.

There is a famous inequality discovered by Golod and Shavarefich that links those numbers above for making a statement whether a (a priori infinite) pro-$p$-group is in fact finite. This is a very beautiful and farreaching result as you can find a lot of applications in Galois theory. (Key words are: class field tower, Hilbert class field = maximal abelian unramified extension)