In this Letter, we give an analytical quantum description of a nonequilibrium polariton Bose-Einstein condensate (BEC) based on the solution of the master equation for the full polariton density matrix in the limit of fast thermalization. We find the density matrix of a nonequilibrium BEC, that takes into account quantum correlations between all polariton states. We show that the formation of BEC is accompanied by the build-up of cross-correlations between the ground state and the excited states reaching their highest values at the condensation threshold. Despite the nonequilibrium nature of polariton systems, we show the average population of polariton states exhibits the Bose-Einstein distribution with an almost zero effective chemical potential above the condensation threshold similar to an equilibrium BEC. We demonstrate that above threshold the effective temperature of polaritons drops below the reservoir temperature.