The adaptive moving mesh algorithm plays a very important role in the numerical solution of singularly perturbed differential equations. The key technology here is the construction of an effective discrete scheme and the corresponding a posteriori error estimation. Based on this, for a class of nonlinear singularly parameterized problems, the stability estimates of continuous solutions and related corollaries are given. Then, a hybrid finite difference scheme is established by using the backward Euler formula and the first-order central finite difference scheme on an arbitrary nonuniform grid, and the stability of the discrete solution is analyzed. Based on this stability estimation and the piecewise linear interpolation technique, an a posterior error estimation of the maximum norm of the mixed finite difference scheme is given. Using the a posterior error estimation, an optimal grid monitor function is selected, and an adaptive grid generation algorithm is designed based on the mesh equidistribution principle. Finally, numerical experiments verify the effectiveness of the adaptive moving mesh algorithm, and the average convergence order of the algorithm can reach the second order. Furthermore, it is shown from the numerical results that the error of the adaptive moving mesh is obviously smaller than that of the Shishkin mesh, and its convergence order is higher than that of the Shishkin mesh.