Nonlinear matrix equation $X-A^{T}X^{-1}A=Q$ has been widely applied to control theory, dynamic programming, interpolation theory and stochastic filtering. In this paper, an equivalent form of this equation is derived, and the Newton's iterative method is applied to solving this equivalent equation. By defining a class of matrix functions which have the property that the matrix sequence generated by the Newton's method to compute its root is the same as that generated by the Newton's method to solve the nonlinear matrix equation, we prove that the matrix sequence generated by the Newton's method to solve the nonlinear matrix equation is included in the closed ball which has an unique solution to the matrix equation. It is also convergent to the unique solution in that closed ball. The error estimate of the approximate solution with the true solution is derived, and a numerical example to illustrate the efficiency of Newton's method is also given.