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. This doctrine is identified with Nestorius (c.386-451), Patriarch of Constantinople, although he himself denied holding this belief. 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 the divinity itself, the Trinity, or one of the persons of the Trinity. Thus, they would say, Jesus the perfect man suffered and died, not the divine second person of the Trinity, for such is an impossible thought -- hence the inference that two "persons" essentially inhabited the one body of Jesus. Nestorius himself argued against calling Mary the "Mother of God" (Theotokos) as the church was beginning to do. He 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 and insisted that Theotokos was an appropriate title for Mary.
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. 
- Nestorianism (basictheology.com)