To complement Adam Saltz's nice answer (+1 BTW): cohomology theories give a way of proving that certain sequences are exact---for instance, of proving that certain maps are surjective (i.e., solving certain equations) or injective. More specifically, and towards your question "What does it bring to the studies involved?", given an exact sequence in an abelian category
$$0 \rightarrow A_1 \rightarrow A_2 \rightarrow A_3 \rightarrow 0$$ and and a left exact functor $F$ (simple example: take the abelian category to be $kG$-modules, where $G$ is a $p$-group and the field $k$ has characteristic $p$, and take $F$ to be the functor of $G$-fixed points), one wants to know under what circumstances the induced map $F(A_2) \rightarrow F(A_3)$ is surjective. Tautologically, cohomology gives a sufficient condition: it's enough to check that the first derived functor on $A_1$ vanishes: $R^1 F (A_1)=0$. This may seem like no real gain. However, one thing that the cohomological machinery provides is a new technique for proving this: try to show, by descending induction, that all the higher cohomology groups $R^i F(A_1)=0$ (here, you need some kind of vanishing theorem that shows they are zero for $i$ large enough, and then you can try to use all the long exact sequences and things like the 5 lemma and its relatives).
A good example of this type of argument to keep in mind is Grothendieck's proof of Zariski's main theorem, e.g. as reproduced in Hartshorne's algebraic geometry book: see the theorem on formal functions and its corollaries.