Nestorius essentially proposed that there were two persons co-existing in Jesus’ head. One of them was the Word or Logos or Second Person of the Trinity: God the Son. The other was an ordinary guy named Jesus who was, as it were, occupied by the Logos as gas fills a balloon. The man Jesus took his humanity from Mary, according to Nestorius. So, said Nestorius, it was blasphemous to call Mary Theotokos (i.e, God-Bearer or Mother of God)–as the Church had done since at least the second century and very likely since apostolic times. Rather, said Nestorius we can only speak of Mary as “Christotokos” or “Christ-Bearer” since the only thing she contributed to the deal was the humanity of the ordinary guy named Jesus who was not God.
This ignited a huge crisis. Why? Because if Nestorius was to be believed, then Jesus is no longer the God-Man, bearing our sins to the cross and rising to give us his divine life. The humanity of Jesus is cut off from the deity of the Logos. The man who died on the cross is not God, but a mere creature occupied by the Logos. Conversely, the God who occupies that man does not share our humanity and therefore can neither die for our sins as a fellow son of Adam, nor share his divine life with us, because he doesn’t even share it with the human bio-envelope named Jesus that he was inhabiting. Indeed, some of Nestorius’ followers thought we could speak of two “Sons of God”: Jesus the man and the Logos. Other Nestorians claimed the “mere human Jesus” had actually earned his own salvation from the Logos by serving as the suffering and dying bio-envelope. It was a formula for spiritual chaos, it was spreading (since the Church at Constantinople was the most powerful and influential of its day), and the Church quickly came to realize she had a problem on her hands that could not be avoided. Nestorius had to be answered.
This the Church did at the Council of Ephesus in a.d. 431. First, she reasserted the basic gospel facts about Jesus by reaffirming he was really and truly God and really and truly human. As the council put it, Jesus was not two persons, but one person with two natures, divine and human. In other words, the Church appealed to Scripture—“The Word was with God, and the Word was God” (John 1:1); “The Word became flesh” (John 1:14); and, “You will conceive in your womb and bear a son, and you shall call his name Jesus” (Luke 1:31)—read within the context of the Church’s longstanding tradition about Jesus.
But the council also realized this reasoning inevitably takes us back to Mary and to the title of Theotokos. The Church’s logic was simple and deeply biblical:
Is Jesus God? (Yes.)
Is Mary his Mother? (Yes.)
Then, said the Church, we ought to call Mary Theotokos or “God Bearer” or “Mother of God,” as we have already been doing for centuries. The title, like all things else pertaining to Mary, was a commentary, not so much on Mary, as on who Jesus is: the one fully human and fully divine Son of God.
In reaction to Nestorius, Monophysites insisted that Christ’s human nature was annihilated by his divine nature (necessitating another ecumenical council, at Chalcedon in 451). Again, the Church reiterated:
We confess that one and the same Christ, Lord, and only-begotten Son, is to be acknowledged in two natures without confusion, change, division or separation.
The Church thus confesses that Jesus is inseparably true God and true man. He is truly the Son of God who, without ceasing to be God and Lord, became a man and our brother. (CCC 469)
 “Hence we are not surprised to find that Diodorus admitted two Sons.” (See the Catholic Encyclopedia on “Nestorius and Nestorianism.”) Available at http://oce.catholic.com/index.php?title=Nestorius+and+Nestorianism as of August 5, 2015.
 “Cassian represents the Constantinopolitan patriarch as teaching that Christ is a mere man (homo solitarius) who merited union with the Divinity as the reward of his Passion.” (See the Catholic Encyclopedia on “Nestorius and Nestorianism.”)
 Council of Chalcedon: DS 302.