This is a Riemann surface.

This image was generated using MATLAB (TM).

This is a projection of the Riemann surface for the cube root function. In this image, the sheets do not actually intersect except at the origin, which is known as a branch point.

Riemann surfaces arise in the study of complex analytic functions: Because every complex number has two square roots, three cube roots, and so on, the cube root function cannot be continuous over all of the complex plane. We call these functions "multiple-valued," and in order to treat them as continuous single-valued functions again, it is necessary to replace their domains by the corresponding Riemann surface. See Complex Analysis by Lars Ahlfors for a short intuitive treatment of this topic (at the "cut-and-paste" level). There are more detailed references, but unfortunately I don't know them offhand.