From what I gather, Nestorius rejected Mary as the Theotokos (which means "God-bearer") and proposed instead that she merely be called, the Christotokos; or rather, "Christ-bearer" because since Mary was a human being---God could never be born of a creature. Hence, (with some details omitted) this would necessarily imply that there were two distinct persons in the Incarnate Christ---one human and one divine accidentally united in one body (in violation of the dogma of homoousios defined at the First Ecumenical Council of Nicea in 325 that God the Father and God the Son are of the same substance).
It is not clear whether or not Nestorius explicitly taught that there exists two natures in Christ, but his doctrine necessarily leads to that conclusion.
Hence, with this heresy, it could neither be said that God was born, nor that He was crucified, nor that He died---and therefore, neither that He rose from the dead. And so, as the Holy Spirit tells us in 1 Cor. 15:17: if Christ be not risen again, your faith is vain, for you are yet in your sins.
The First Council of Ephesus in 431 condemned Nestorius' teachings as heretical and deposed him from his See (Nestorius was the patriarch of Constantinople at the time.)
Was Nestorius ever formally excommunicated by the Catholic Church?
I have now noticed that this question seems to be somewhat connected with the one posed here: Proof that Cyril of Alexandria is a Saint in the Roman Catholic Church?, for Catholic Encyclopedia in the entry for Pope St. Celestine I says (regarding St. Cyril of Alexandria):
Cyril having found Nestorius openly professing his heresy sent a full account to Celestine, who in a Roman synod (430), having solemnly condemned the errors of Nestorius, now ordered Cyril in his name to proceed against Nestorius, who was to be excommunicated and deposed unless within ten days he should have made in writing a solemn retractation of his errors. In letters written the same day to Nestorius, to the clergy and people of Constantinople, and to John of Antioch, Juvenal of Jerusalem, Rufus of Thessalonica, and Flavian of Philippi, Celestine announces the sentence passed upon Nestorius and the commission given to Cyril to execute it. At the same time he restored all who had been excommunicated or deprived by Nestorius. Cyril forwarded the papal sentence and his own anathema to Nestorius.
Nevertheless, it is not clear to me whether or not Nestorius had been officially excommunicated.