Monophysitism is an error concerning the nature of Christ that asserts Jesus had only one nature and not two as is taught in the correct doctrine of the hypostatic union: Jesus is both God and man in one person. In monophysitism, the single nature was divine and not human. It is sometimes referred to as Eutychianism--after Eutyches 378-452, but there are slight differences. Monophysitism arose out of a reaction against Nestorianism which taught Jesus was two distinct persons instead of one. Its roots can even be traced back to Apollinarianism which taught that the divine nature of Christ overtook and replaced the human one.
Monophysitism was confined mainly to the Eastern church and had little influence in the West. In 451, the Council of Chalcedon attempted to establish a common ground between the monophysitists and the orthodox, but it did not work and divisions arose in the Eastern church which eventually excommunicated the monophysitists in the 6th century.
The denial of the human nature of Christ is a denial of the true incarnation of the Word as a man. Without a true incarnation there can be no atonement of sin for mankind since it was not then a true man who died for our sins.
It was condemned as heresy at the Sixth Ecumenical Council in 680-681.