The nonlinear Schrodinger equation is a nonlinear partial differential equation and integrable equation that play an essential role in many branches of physics as nonrelativistic quantum mechanics, acoustics, and optics. In this work, motivated by the ideas of Ablowitz and Musslimani, we successfully obtain a two-dimensional nonlocal nonlinear Schrodinger equation where the nonlocality consists of reverse time fields as factors in the nonlinear terms. The nonlocal nonlinear Schrodinger equation admits a great number of good properties that the classical nonlinear Schrodinger equation possesses, e.g. PT-symmetric, admitting Lax-pair, and infinitely many conservation laws. We apply the Darboux transformation method to the two-dimensional nonlinear Schrodinger equation. The idea of this method is having a Lax representation, one can obtain various kinds of solutions of the Nth order with a spectral parameter. The exact solutions and graphical representation of obtained solutions are derived.