“Genus” has number of distinct but compatible definitions.

In topology, if $S$ is an orientable surface, its genus $g(S)$ is the number of “handles” it has. More precisely, from the classification of surfaces, we know that any orientable surface is a sphere, or the connected sum of $n$ tori. We say the sphere has genus 0, and that the connected sum of $n$ tori has genus $n$ (alternatively, genus is additive with respect to connected sum, and the genus of a torus is 1). Also, $g(S)=1-\chi(S)/2$ where $\chi(S)$ is the Euler characteristic of $S$.

In algebraic geometry, the genus of a smooth projective curve $X$ over a field $k$ is the dimension over $k$ of the vector space $\Omega^{1}(X)$ of global regular differentials on $X$. Recall that a smooth complex curve is also a Riemann surface, and hence topologically a surface. In this case, the two definitions of genus coincide.