The genus is a topological invariant of surfaces: that is, a quantity associated with a surface that does not change when the surface is continuously deformed. Roughly speaking, it corresponds to the number of holes of that surface, so a sphere has genus 0, a torus has genus 1, a pretzel shape (that is, the surface of a blown-up figure eight) has genus 2, and so on. If one triangulates an orientable surface and counts the vertices, edges, and faces in the triangulation, denoting their numbers by V, E, and F, respectively, then the Euler characteristic is defined to be V - E + F. It can be shown that if g is the genus and X is the Euler characteristic, then X = 2 - 2g. See [I.4 §2.2] for a fuller discussion.
A famous result of POINCARÉ [VI.61] states that for every nonnegative integer g there is precisely one orientable surface of genus g. (Moreover, genus can also be defined for nonorientable surfaces, where a similar result holds.) See DIFFERENTIAL TOPOLOGY [IV.7 §2.3] for more about this theorem.
One can associate an orientable surface, and therefore a genus, with a smooth algebraic curve. An ELLIPTIC CURVE [III.21] can be defined as a smooth curve of genus 1. See ALGEBRAIC GEOMETRY [IV.4 §10] for more details.