Genus

The dodecahedron (left) has no holes and is genus 0. The polyhedron on the right has a hole through it and is genus 1. Genus is a way of classifying the surfaces of polyhedra. For orientable polyhedra it is a mathematically precise measure of the number of holes the polyhedron has.

Topological definition
On a connected 2-dimensional surface the genus is the maximum number of loop cuts that can be made without disconnecting the surface. This effectively measures the number of holes on a surface because a hole allows you to make a loop that passes through it, while the other side of the loop keeps the surface connected. Each hole added gives an additional possible cut.

For example if you cut any loop in a sphere it divides the sphere into two disjoint sections. Thus the sphere has genus 0. A torus however can be cut without disconnecting it but only once. It has genus 1.

The genus of a polyhedron is simply the genus of its surface. For example all convex polyhedra are homeomorphic to a sphere and have genus 0.

Euler characteristic
Genus can also be defined in terms of the Euler characteristic of a polyhedron. For an orientable surface the genus is $$1-\frac{\chi}2$$; for a non-orientable surface the genus is $$2-\chi$$.

This definition only works for finite polyhedra. The topological definition works for infinite polyhedra as well.

Higher dimensions
The genus of a surface is not defined for polytopes whose rank is not 3. A more general version of genus, the homology group exists and is defined for higher dimensional surfaces.