The way I understand it, Christ is one Person (i.e. there is only one Christ, no other, no second Christ), Who, after the Incarnation, unites in Himself two natures or "ousias," Divine and human. So, when Nestorians in the first half of the 5th century A.D. claimed that it is not right to refer to what the Theotokos gave birth to as "God," because God cannot be born by a woman, the Church said, yes, it is right, because the Person Who was given birth by Her was simultaneously God and man.
Perhaps the best way to refer to the Person of Christ is "God-Man," or God AND man.
Sorry if this is too "lay" - I am sure our Fathers who have a proper theological education will answer you better.