There is a mathematical definition of genus in general and of elliptic genus in particular, which you may or may not find enlightning. It says something like that a genus is an assignment of some quantity to a manifold, such that the quantity depends suitably only on the cobordims class of the manifold and such that the disjoint union of two manifolds is taken to the sum and the product of two manifolds to the product of that quantity.

So then you may ask: is there a physical interpretation of this? Where would such genera appear in physics? And here the striking answer is: at least the important genera turn out to be the assignments that take a manifold, regard it as the target spacetime in which some spinning particle or superparticle or spinning string or superstring or some other brane propagates, and then assigns the quantity which is the partition function of the worldvolume theory of that little brane propagating in that manifold.

This was Edward Witten's big insight, part of what won his Fields medal, regarding the Ochanine elliptic genus, which he effectively understood to be the partition function of the type II superstring. Then he checked what the partition function of the heterotic superstring would give and found this way a new genus, now named after him: the Witten genus.

Since then, genera are being identified all over the place as partition functions of supersymmetric QFTs. One may also think of these are being the indices of the supercharge and hence often they are called indices. Check out this table.