Yes. <a href="http://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theorem">Wagner's Conjecture/Robertson and Seymour's Theorem</a> says that any graph family closed under taking minors can be defined by specifying a finite list of forbidden minors. For any surface $S$, the graphs embeddable $S$ without crossing edges forms a family closed under taking minors. 

I haven't looked carefully at it but <a href="http://cornellmath.wordpress.com/2007/07/04/graph-minor-theory-part-3/">Jim Belk's introduction to graph minor theory</a> seems good. On the linked page he mentions the following facts: the projective plane has 35 forbidden minors, the number for the torus is in the hundreds (the precise number/collection is not known), and in general the number of forbidden minors grows exponentially with the genus.