An elliptic curve, being a curve, is naturally a geometric object, but it is also an algebraic object, because its points have a natural group structure. Elliptic curves have long been a testing ground for number theorists, who are especially interested in the points on elliptic curves having coordinates that are rational numbers or integers. In this talk I will explain what an elliptic curve is, both geometrically and algebraically, describe important theorems of Mordell and Siegel that classify the rational and integral points on elliptic curves, and conclude with a fascinating conjecture of Serge Lang that relates these two fundamental results.