Even if you're only interested in number theory applications of studying elliptic curves, you first have to study their geometry (which is interesting in its own right!). I will discuss what elliptic curves are and how the points on them form a group. Given time, I will also discuss singular elliptic curves, the (classical) theory of complex elliptic curves, and finding rational points on elliptic curves.
The talk should be accessible to everybody (elementary algebra required).