Well you asked ...
Nestorianism is something of an anachronistic catch-all that covers an emerging position in Syria around the 6th-7th centuries.
The 'simple' answer is Nestorius was a Patriarch of Constantinople who tried to resolve a theological dispute in Christology. One the one side, there was the view that in the Incarnation, God was born as a man. The opponents argued that God, by virtue of His eternal nature, cannot be said to have been born. The 'simple' argument was over the title 'Theotokos' which was coming into use with reference to the Virgin Mary. The literal translation is something like 'God-bearer' but it's not quite so simple as there are other Greek terms for both 'God-bearer' and its more popular understanding, 'Mother of God' ... so it's open to interpretation and, as history shows, there are those who read it as implying that if Mary is the mother of God, then she herself is also a God.
But it gets more complex.
The Council of Nicaea (325) declared that Christ was divine (homoousios, consubstantial, of one being or essence, with the Father) and human (was incarnate and became man). In the fifth century controversy arose between the schools of Antioch and Alexandria about how divinity and humanity existed in Christ.
Cyril of Alexandria argued with the Antiochene Nestorius at the Council of Ephesus (431), insisting on the formula "one physis (nature) of the Incarnate Word", claiming that Nestorius favoured a formula of two physeis, two natures and two hypostases (individualities). Nestorius lost, and was dismissed as Patriarch and exiled.
The Council of Chalcedon (451) defined that in Christ there were two physeis (natures) united in one hypostasis in the prosopon (person) of Christ.
Those who insisted on the "one physis" formula were referred to as Monophysites, while those who accepted the Chalcedonian "two natures" definition were called Dyophysites, but this latter term obviously also applied to Nestorians, who held two hypostases, as opposed to Chalcedon's one hypostasis.
(We can ignore the Acephali, the Agnoetae, Themistians or Agnosticists, and Aphthartodocetae, Phantasiasts or Julianists as being peripheral)
Apollinarians or Apollinarists (Apollinaris/Apollinarius, Bishop of Laodicea, died 382) proposed that Jesus had a human body but a divine soul/mind.
Miaphysites are often labelled Monophysites, a label that they reject, basing their theology on the formula of Cyril of Alexandria (376-444) that spoke of mia (one) physis, not of a mone (lone) physis. This is the doctrine of the Oriental Orthodox Churches, which separated from the mainline after Chalcedon. They preach one hypostasis of Christ is one physis (nature), a nature fully divine and fully human.
Severians (Severus the Great of Antioch, aka Severus of Gaza, Patriarch of Antioch, and head of the Syriac Orthodox Church from 512-538). They accept the reality of Christ's human nature but argued that, since a single person has a single nature and Christ was one person, not two, he had only a single nature. They agreed in substance with the Definition of Chalcedon, but not with the wording, and thus are known also as 'verbal Monophysites'.
Eutychians (Eutyches of Constantinople c. 380– 456) taught that Jesus had only one nature, but that it was a union of the divine and human, albeit not equal, since what is divine is infinitely larger than what is human. The humanity is absorbed by and transmuted into the divinity, as a drop of honey, mixing with the water of the sea, vanishes. They are known as 'ontological Monophysites'.
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Nestorianism, if what we understand was his position, is a radical dyophysitism (two physeis and two hypostases) and differs from the orthodox dyophysitism (two physeis, one hypostasis) of Chalcedon.
After the condemnation, some supporters of Nestorius relocated to Syria, then in the Sasanian Empire, where they were affiliated with the local Christian community, known as the Church of the East. This had emerged in the early 5th century and declared itself independent in 424.
Headed by the Patriarch of the East seated in Seleucia-Ctesiphon, it traced its origin to Thomas the Apostle in the first century. The Church of the East was counted as orthodox, sharing communion with Rome, but refused to condemn Nestorius and thus stood accused of Nestorianism, and called the Nestorian Church by its detractors.
Specific doctrinal views emerged over time, their final form in the teachings of the East Syrian theologian Babai the Great (d. 628) who used the Syriac term qnome (Syriac of the Greek hypostasis) as a designation for dual (divine and human) substances within one prosopon (person or hypostasis) of Christ.
Opponents labeled him "Nestorian", thus creating the practice of misnaming the Church of the East as Nestorian. Nestorius was venerated as a saint in the Church of the East, but recent studies the label has been criticised as improper and misleading. As a consequence, the use of Nestorian label in scholarly literature, and also in the field of inter-denominational relations, is gradually being reduced to its primary meaning, focused on the original teachings of Nestorius, rather than the doctrinal position of the Church of the East.
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Sometime around the eighth century East Syrian Dyophysite Christians began to refer to themselves as ‘Nestorian’, despite its pejorative connotations. Shahdost, the Bishop of Tirhan, said:
"... it is evident that it was not ourselves (who) have turned aside from the foundation of the faith, we who, in the orthodox way, believe that in Christ, are two natures and two hypostases... Severus and his disciples have turned aside, they who confess Christ to be one nature and one hypostasis. But we, the Nestorian Christians, cry with John the Evangelist: 'The Word became flesh and dwelt among us'"
Historical documents evidence a delegation of Persian theologians in a discussion held before Justinian, talking of a two-qnona Christology was already formulated about 561/563. But the bishops had not yet formulated a creed according to that formula.
Henana of Adiabene was the Director of the theological school of Nisibis (c 571-610), and he introduced a one-qnome Christology, thus laying the foundation for schism, which caused serious troubles in the East Syrian Church.
Babai the Great, Abbot of the Great Monastery of lzla north-cast of Nisibis (604--628), championed the two-qnoma theology in fierce polemical attacks on Henana and the one-qnome school. He regarded Henana as an innovator, breaking with Apostolic and Patristic tradition. That was not entirely the case, but so effective was Babai that the freedom to express an alternative view ceased to exist.
The arguments rumbled on, disputes between what was emerging as orthodox two-nature Nestorianism, and one-nature miaphysitism the Oriental Orthodox Church.
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After the Muslim conquest of the Sasanian Empire in 644, the newly established Caliphate recognised the Church of the East as an official minority group under its Patriarch. Forbidden to proselytise at home, they established dioceses in India (the Saint Thomas Christians), Central Asia, China, Sri Lanka and possibly Tibet.
Nestorian Christians made substantial contributions to the Islamic Umayyad and Abbasid Caliphates, translating the works of the Greek philosophers to Syriac and Arabic. They made their own contributions to philosophy, science and theology, and Assyrian Christians became personal physicians to the Abbasid Caliphs.
