particular kinds of riemann surfaces? there aren't that many kinds, at least topologically. complex-analytically, there are more kinds, but not being complex analysts, one doesn't care so much about hte distinction. topologically, they're all spheres with N handles glued on to them, if they're closed. if open, they may have some puncture wounds.
the string lives in another space of dimension 10. six of those dimensions are curled up small. so maybe it was those spaces you were thinking of. the shape of the curled up space determines the nature of what the particle physics looks like at low energies (i.e. from far away) - what forces are present in it. physical consistency of the theory requires that a particular curvature invariant of the six dimensional space vanish. such spaces are called Calabi-Yau manifolds. An important example is the K3 surface…..
anyway, the Calabi-Yau is part of the space that the string moves and wiggles in.
(some good coverage is in 'The Elegant Universe', by Brian Greene.)
K3 surfaces, by Paul Aspinwall - http://xxx.lanl.gov/abs/hep-th/9611137
see: Membrane Theory