Compact Riemann surface
Riemann surfaces are generally classified first into the compact (those that are closed manifolds) and the open (the rest, which from the point of view of complex analysis are very different, being for example Stein manifolds).
As algebraic curves
Every compact Riemann surface C that is a connected space can be represented as an algebraic curve defined over the complex number field. More precisely, the meromorphic functions on C make up the function field F on the corresponding curve; F is a field extension of the complex numbers of transcendence degree equal to 1. It can in fact be generated by two functions f and g. This is a structural result on the meromorphic functions: there are enough in the sense of separating out the points of C, and any two are algebraically dependent. These facts were known in the nineteenth century (see GAGA for more in this direction).
A general compact Riemann surface is therefore a finite disjoint union of complex (non-singular) algebraic curves.