A Riemann surface is a complex analytical variety of dimension 1; put more simply, it is a surface that locally has the same properties as a disc in the complex plane. They were devised and studied by Riemann to overcome the many-to-many character of certain functions in the complex domain (logarithms, roots etc.). Riemann surfaces are often seen as wrappings (branched) of the complex plane, i.e. as a particular collage of several "sheets". They often admit multiple interconnections that are tricky to represent in three dimensions.
The Riemann surface associated with the complex logarithm: this is a wrapping of the plane with an infinity of sheets, admitting a branching point at the origin.