First, an elliptic curve is not an ellipse. The name reflects a connection with elliptic functions. Elliptic curves are nonsingular plane curve defined by a certain class of cubic equations. Each elliptic curve gives rise to a mathematical object called an L-series. Since a complete L-series is an infinite product, it is hard to use in further calculations.
Modular form is an analytical function and it offer a short cut to the analysis of L-series.
The connection between the elliptic curves and modular forms is that modular forms also give rise to L-series analogous to those coming from elliptic curves. In 1985, the German Gerhard Frey drew attention to this equation by stating that the elliptic curve, which is now often called the Frey curve, could not be modular. However, he was not able to give a rigorous proof. A year later, Kenneth A. Ribet of UC Berkeley supplied the proof. The main point of Wiles' proof is the connections between the elliptic curves and the modular forms.