The notion of genus has several equivalent definitions. Through algebraic geometry has many generalizations that you're probably not interested in. Genus, is something that you can intuitively understand in $\mathbb{R}^3$, where the 2-dimensional manifolds (i.e. surfaces )do exist. That is, the number of holes such a surface has, for instance for a torus the genus is $g=1$, connected sum of two tori is $g=2$, so on so forth. Apparently you can have genus zero (2-dimensional sphere is such an instance), where no holes whatsoever show up. Because elliptic curves are complex manifolds of dimension one (complex dimension), that means that have dimension 2 over $\mathbb{R}$ (keep in my mind, we always talk about non-singular objects) and we can transfer this intuitive definition to a non-singular curve over $\mathbb{C}$. Now, since you already know that they have genus 1, that means its real analogue must be what we call torus in $\mathbb{R}^3$. Indeed that's certainly true since you can prove that for an elliptic curve $E$, there is an isomoprhism of the form $E \cong \mathbb{C}/{\Lambda}$, where $\Lambda$ a lattice in the complex numbers. I haven't done a thorough explantion for any of the above, because the usual problem is when an OP asks something no one has idea of their background and that creates problems (the latter comment is just in case @Rene Schipperus intimidate you :P). I did all the above introduction to give maybe some intuition behind the formal definition that you already know (btw in that context the formal definition I mean is this one described here in section orientable surfaces https://en.wikipedia.org/wiki/Genus_(mathematics) and I believe is quite tangible).
The above maybe are not enough, or too many to answer clearly your question because I don't know your background in algebraic/differential geometry. Though, as I have already mentioned above in comments, in case you understand what Hartshorne's book mean by abstract variety there is plenty of information in there to help you out.
For any possible questions do let me know.
