A Riemann surface $X$ is a connected complex manifold of complex dimension one. A homogeneous space is a manifold with a transitive smooth action of a Lie group. I guess there must be a classification of those Riemann surfaces which are also homogeneous spaces . . . however, I can't seem to find it.
If you only care about compact (oriented) surfaces, then it is easy to see that the only homogeneous examples are the sphere $S^2=\mathbb C P^1$ and the torus $T^2=S^1\times S^1$. Indeed, if $M$ is a compact homogeneous space, then its Euler characteristic is $\chi(M)\geq0$; so for an oriented surface this implies genus $\leq1$, as Pierre Dubois stated in his comment. Perhaps the references mentioned here can be helpful to you: How do you see that higher genus surfaces are not homogeneous?