The finite difference method is one of the most widely used numerical techniques for solving the problem of the differential equations. Its fundamental idea is to approximate the original differential equations and their corresponding boundary conditions with discrete difference equations involving a finite number of unknowns. Then the solution of these difference equations serves as an approximation to the solution of the original differential equations. The two-point boundary value problem of an ordinary differential equation can be interpreted as a boundary value problem of one-dimensional elliptic equation. In this chapter, we explore the finite difference solution to a model problem and introduce several key concepts in the numerical solution for differential equations, including the maximum principle, the energy method, and Richardson extrapolation.