The purpose of this project work is to understand the Euler-Lagrange Equation and to show the application of Euler-Lagrange equation on solving variational and real life problems. The calculus of variations addresses the need to optimize certain quantities over a set of functions. In order to identify those functions which are extremals of functionals, we establish the Euler- Lagrange equations. Euler-Lagrange equation is a second order partial differential equation whose solutions are functions for which the given functional is stationary. Euler-Lagrange equation is useful for solving optimization (extremization) problems; such as minimizing surface area of revolution, finding the shortest path joining two distinct points, finding the path joining two distinct points that takes shortest time, finding the curve that encloses the largest area with fixed perimeter etc., that are analyzed by the concept of calculus of variation.