Here is a writeup I made for my undergraduate Algebraic Geometry course (Math 106) at UCSD. It is a brief overview into the theory of elliptic curves, some motivation behind it, and some potential applications (primarily to solving Diophantine equations and to the Congruent Number Problem)! I think there may be a few typos / issues with it, but hopefully someone may find it interesting in the future. One of these days, I might turn it into a blog post.