Nestorianism
Nestorianism is basically the doctrine that Jesus existed as two persons, the man Jesus and the divine Son of God, rather than as a unified person.
History
Nestorianism is identified with Nestorius (c.AD 386-451), Patriarch of Constantinople. This view of Christ was condemned at the Council of Ephesus in 431, and the conflict over this view led to the Nestorian schism, separating the Assyrian Church of the East from the Byzantine Church.
The motivation for this view was an aversion to the idea that "God" suffered and died on the cross, be it God himself, or one of the persons of the Trinity. Thus, Nestorians would say, Jesus the perfect man suffered and died, not the divine Son of God, for such is an impossible thought -- hence the inference that two "persons" essentially inhabited the one body of Jesus. Nestorius held that Mary was the mother of Christ only in respect to His humanity. The council at Ephesus (431) accused Nestorius of the heresy of teaching "two persons" in Christ.
The problem with Nestorianism is that it threatens the atonement. If Jesus is two persons, then which one died on the cross? If it was the "human person" then the atonement is not of divine quality and thereby insufficient to cleanse us of our sins.