Nestorianism is the error that Jesus is two distinct persons. The heresy is named after Nestorius, who was born in Syria and died in 451 AD, who advocated this doctrine. Nestorius was a monk who became the Patriarch of Constantinople and he repudiated the Marian title "Mother of God." He held that Mary was the mother of Christ only in respect to His humanity. The council of Ephesus was convened in 431 to address the issue and pronounced that Jesus was one person in two distinct and inseparable natures: divine and human.
Nestorius was deposed as Patriarch and sent to Antioch, then Arabia, and then Egypt. Nestorianism survived until around 1300.
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.