In this paper, the uniqueness of the Moore-Penrose inverse of a matrix is proved and an algorithm for solving the Moore-Penrose inverse of a matrix is proposed. First of all, the Moore-Penrose inverse of the matrix is reversed to the solution of a matrix equations with three matrix variables. Then a modified conjugate gradient algorithm (MCG algorithm) is established to solve the matrix equations. Moreover, the properties and convergence of the MCG algorithm are proved. For any given initial matrix, we can obtain the Moore-Penrose inverse of the matrix after finite iterative steps. Finally, several numerical examples are given to prove that MCG algorithm has high computational efficiency for solving the Moore-Penrose inverse of a matrix.