Where can I find a concrete description of mapping class group of surfaces?  I know the mapping class group of the torus is $SL(2, \mathbb{Z})$.  Perhaps, there is a simple description for the sphere with punctures or the torus with punctures.  Also, I would appreciate any literature reference for an arbitrary surface of genus g with n punctures.  

Mapping class groups come up in my reading about billiards and the geodesic flow on flat surfaces.  I wonder: the moduli space of complex structures on the torus is $\mathbb{H}/SL(2, \mathbb{Z})$, is it a coincidence the mapping class group appears here?