$H^1(X,\mathcal G)$ where $X$ is any topological space, and $\mathcal G$ any sheaf of groups, classifies $\mathcal G$-torsors on $X$.  A $G$-torsor is a sheaf of sets with a $\mathcal G$-action on $X$ which is isomorphic to the sheaf $G$ on small enough open sets.

If $\mathcal G$ is the set of locally constant functions valued is a group $G$, then a $\mathcal G$-torsor is really just an $G$-local system (this is essentially by definition), so for any group this result holds.

**EDIT**: Paul's answer made me realize I hadn't proceeded to then point out that a local system is the same thing as a map $\pi_1(X)\to G$ up to conjugacy, by taking the monodromy along paths (which requires choosing base points, hence the conjugacy).  I wasn't trying to be mysterious; I just have this so ingrained in my head that I forgot it was worth mentioning.

You have to careful here, because when $G$ is a topological group, it's very tempting to think about the sheaf of continuous functions to $G$ on $X$; that's the sheaf of groups whose torsors are principal bundles. This will only agree with the above if $G$ is discrete (hence Angelo's answer).