Abstract: This talk will be an introduction to elliptic curves and is meant for a general audience. During the talk I'll explain what elliptic curves are and why they are useful objects of study. The first half of the talk will be spent defining elliptic curves over Q and F_p and surveying some of the major theorems related to the structure of their groups. The second half of the talk will be devoted to the congruent number problem. An integer n is said to be a congruent number if there exists a right triangle with rational side lengths whose area is n. For a very long time, little was known about the classification of congruent numbers. In fact, it is still not even known whether or not they constitute a computable set. In 1983, Tunnel proved a remarkable theorem relating congruent numbers to the values assumed by certain quadratic forms. The key to the proof of his theorem was elliptic curves, and it is by outlining the logical structure of his argument that we'll see some of the more advanced topics related to elliptic curves (e.g. the Birch-Swinnerton-Dyer Conjecture).