The Council of Ephesus, convened on June 22, 431, met to discuss Nestorius Bishop of Constantinople.
John of the Golden Mouth, aka Chrysostom, bishop of Constantinople, who died in A.D. 407, was known for his oratorical skill. In 428, when the Emperor Theodosius II established Nestorius as Patriarch of Constantinople, it was hoped that Nestorius, who was also known for his sermons, would serve just as well. At first, all seemed right. Nestorius got right on the task of issuing edicts against heresies -- notably the Arian heresy, although he didn't seem bothered by the followers of Pelagius, whose heresy became popular in Britain, Palestine, and Carthage.
This was the time when Christian doctrine was being decided. It was also during the time of the split Empire, when there was a ruler in the West and another one in the East, and when the Empire was linguistically divided into Latin and Greek. Nestorius and Theodosius were in the Greek-speaking East.
Nestorius and his followers questioned the idea that the Virgin Mary was Theotokos. Instead, they called her a Christotokos. The Bishop of Alexandria, Cyril, heard about this and used it for a power play to wrest power from the East. The patriarch of Rome, or pope, Celestine I, supported Cyril against Nestorius. Presumably these men also sincerely believed that Mary was the Theotokos and that Nestorius was theologically at fault.
Poor Nestorius and his followers thought their position was just as strong, so they went along with the idea of invoking another Council to decide this and other matters. The Council of Ephesus was therefore held in 431. In the end, the emperor changed his allegiance to Cyril, and so a disgraced Nestorius went into exile in Egypt where he established Nestorian churches.
What was the big deal about the Theotokos controversy? Today it is hard to understand because most of us don't pay attention to the finer points of theology. Probably a good number of devout Western Christians are Nestorians.
Theotokos* is the person who gives birth to a god. Christotokos, the term the Nestorians preferred, is the person who gives birth to a christos (anointed one, i.e., the messiah). The Theotokos could be giving birth to a god who is an anointed one/messiah, but the Christotokos is giving birth to a human savior. Thus the Theotokos is in a sense more inclusive. The child of a Theotokos could be human and divine, where the child of a Christotokos would be human. That means that a Christotokos Mary only gave birth to the human aspect of Jesus. Making her the mother of a god puts her in a very special position, worthy of reverence as the Blessed Virgin Mary. Nestorius' position made Christ a sort of split personality with two separate essences. The doctrine that became dominant is that Jesus and the Son of God are one completely unified person. At least this is my understanding.
Please add to the comments to clarify further.
*Theotokos is sometimes translated 'god-bearer', like the related doryphoros 'spear-bearer'. There is a distinction between tokos and phoros that is hard to translate. In Greek mythology, Semele is a theophoros, but she dies before giving birth. Zeus sews the fetal Dionysus in his thigh and then gives birth to him, making Zeus a theophoros and a theotokos.