The christological controversies of the fifth century were arguments about how to say something all parties agreed on: that Jesus Christ is fully God and fully human, one subject and not two.
Nestorius was condemned at Ephesus in 431 for a way of speaking that appeared to make Christ two persons in loose association. Cyril of Alexandria, who led that condemnation, used a formula he believed came from Athanasius: one incarnate nature of God the Word. Eutyches was then condemned in 448 for taking the formula to mean that Christ's humanity was absorbed into his divinity like a drop of honey in the sea, so that he was no longer consubstantial with us. That position is properly called monophysitism, and every party in the dispute rejected it.
The Council of Chalcedon in 451 defined Christ as one person in two natures, divine and human, without confusion, change, division or separation. The definition was framed with Eutyches in view, but it used the word nature in a sense that Cyril's followers could not accept, since for them one nature was the guarantee of a single subject. The churches of Egypt, Syria and Armenia refused to receive the council, holding that its language divided Christ and departed from Cyril, and that the imperial enforcement that followed was coercion.
Miaphysitism, the resulting position, holds that divinity and humanity are united in one incarnate nature, without mixture, confusion, alteration or separation. The humanity is complete and remains what it is; the union is real and permanent; there is one subject. Oriental Orthodox theologians point out that they use the same four adverbs Chalcedon does, and they reject the label monophysite as a description of Eutyches' error rather than of their own faith.