Any connected closed (i.e. compact) surface is homeomorphic to precisely one of a list of standard surfaces. The orientable standard surfaces are the tori of differing genus g≥0; the non-orientable standard surfaces are the sphere with k>0 Möbius strips sewn in. The standard surfaces are topologically distinct on account of the orientability (or not) and their Euler characteristic.

http://www.maths.ed.ac.uk/~aar/surgery/zeeman.pdf A well-illustrated description of some types of surfaces.