Nestorianism asserts that Christ existed as two separate persons, divine and human, in the same earthly body. This is generally considered a heresy, as it contradicts to the belief of most Christians that Christ was a human incarnation of God.
Nestorius (386-451 AD), who was the Archbishop of Constantinople, was the chief proponent of this doctrine, which led to the Nestorian Schism, after the condemnation of this belief by the Council of Epheseus. This schism separated the Assyrian Church from the rest of the Christian community, as they refused to renounce Nestor.
A typical Nestorian would not accept the statement "Mary was the mother of God", because in their eyes, she only gave birth to Christ the man. Similarly, they would contend that God, the Word, did not suffer on the cross, while Jesus the man died.
The Nestorians controlled the Church in Persia for centuries.