Eutychianism is a Monophysitism heresy. This sect teaches that the human nature of Christ was overcome by the divine aspects, or that His human nature was unlike the rest of humanity—it was often stressed that the unity of Christ’s nature was to such an extent that His divinity consumed His humanity. Eutyches (for whom this heresy is named) said that Christ was of two natures, but not in two natures (i.e. that he was homoousian with the Father, but not homoousian with His humanity). Eutychianism was condemned at the Council of Chalcedon in 451, where the Chalcedonian Creed was written to counter the Eutychian and Miaphysitism heresies. This reaction by the Council lead to the schism with the Oriental Orthodoxy Church.
Miaphysitism is a moderate form of Monophysitism (teaching that both natures are merged into one, rather than that the divine subsumes the human). After the 451 Council of Chalcedon, where the two natures of Christ were affirmed, this belief caused a split with the Oriental Orthodoxy Church (other issues, including political, were also at play), who still practice Miaphysitism today (although they object to that term being used to describe them).