What is a short exact sequence telling me? Let's take a short exact sequence of groups
$$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$$
I understand what it says: the image of each homomorphism is the kernel of the next one, so the one between $A$ and $B$ is injective and the one between $B$ and $C$ is surjective. I get it. But other than being a sort of curiosity, what is it really telling me?

This set up will tell you that, essentially, $C=B/A$. Think of the first isomorphism theorem.
In some cases, you get that $B=A\oplus C$ (see Wikipedia: Splitting lemma).
