First case: Complex number. Over $\mathbb C$ the structure as an abstract group is $\mathbb S^1 \oplus \mathbb S^1$ where $\mathbb S^1$ is the circle, i.e., $\mathbb R/\mathbb Z$. This follows as Robin Chapman denotes below, i.e., it is a complex torus in the form $\mathbb C/\Lambda$ where $\Lambda$ is a lattice in $\mathbb C$.
Let $K$ be an algebraically closed field of char $p$. If $n$ is prime to $p$, the the $n$-torsion is $\mathbb{Z}/n\mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$. The $p^e$-torsion could be either trivial for all $e$, or $\mathbb{Z}/p^e\mathbb{Z}$ for all $e$.
Over a non-algebraically closed-field, this is going to be much more complicated. I will try to give just an introduction.
Over $\mathbb Q$ and number fields: Over $\mathbb Q$ or a number field, it is finitely generated by the Mordell-Weil theorem. So it has a torsion part and free part. The free part could be arbitrarily large.
The Mordell-Weil theorem is in fact true for arbitrary finitely generated fields. This is due to Néron.
Over $\mathbb Q$, The torsion part has exactly $15$ possibilities according to the theorem of Mazur. Over number fields, this had been generalized that the torsion part is uniformly bounded.
Over finite fields, the torsion group would be $\mathbb{Z}/n\mathbb{Z} \oplus \mathbb{Z}/m\mathbb{Z}$ where $n |m$. There is no free part.
And it could go on like this. Please have a look at Silverman's "Advanced topics in the Arithmetic of Elliptic Curves" for elliptic curves over real numbers, $p$-adic numbers, function fields, etc..
