A finite difference approximation to a singularly perturbed 1-D parabolic equation is constructed and studied. The nonlinear scheme is based on the method of lines and uses the technique of the exact difference schemes in space direction. The construction uses a special nonuniform grid, locally refined in the subdomain of the boundary layer, so that the convergence is uniform with respect to the small parameter. At the end, an interesting application of this approach for construction and study of difference approximation to a parabolic equation with boundary layer and moving boundary is given.