In the West the feast is called the Assumption, for both Roman Catholics and Orthodox believe that Mary was assumed bodily into heaven. There is, of course, no mention of this in the New Testament. The story comes from apocryphal sources but it has a secure place in the memory and Tradition of the Church for all that. We believe it, because it accords with the experience of the Church and her theology of Incarnation and Resurrection.
The Old Testament tells us that Enoch and Elijah were assumed bodily into heaven. We believe, however, that Mary, being sanctified by grace was chosen because of her purity to be the Mother of God. This purity and God-bearing in her womb, together with her total dedication to her Son fitted her in this life to share immediately in the resurrection of her Son and so she, like Enoch and Elijah before her, was assumed into heaven. This Assumption attests to the fact that she has been deified, that is, her humanity has been glorified by God and united with him. As followers of Jesus Christ, we are promised that we also shall all be deified, achieving the likeness as well as the image of God, and receive Resurrection bodies (though for most of us all this will happen beyond this life). The Mother of God is therefore our example and inspiration. She is the first to receive her crown of glory as Queen in heaven, but no less shall we with her be kings and queens also by the saving work of God.
Although Orthodox Christians believe in the Assumption of the Theotokos, it has never been made into a dogma of the Church (as it has in the Roman Catholic Church). This is because the Dormition or Assumption is something that follows on from the resurrection of Christ but is not part of our public preaching of salvation by Christ, which dogma can only can be.
Indeed apart from "Theotokos" or "God-bearer" which is a title defending the Incarnation, the Orthodox Church has generally avoided formulating doctrines about the Mother of God. "Theotokos" requires that we believe that she is the Virgin Mother of our Lord and God and Saviour, Jesus Christ, who is both God and Man. The Dormition is perhaps best summed up in one of the great hymns sung at Vespers for today (it is the last idiomelon of the aposticha):
"When you were translated to him who was born of you in an inexplicable manner, O Virgin Theotokos, there were present James, the brother of the Lord and first of the Chief Priests, and Peter, the honoured head and leader of theologians, with the rest of the divine rank of Apostles, clearly uttering divine words, praising the amazing divine mystery, the mystery of the dispensation of Christ God, and with joy preparing your body which was the God-receiving originator of life, O most glorified one, while the most holy angels looked on from on high, struck with astonishment and surprise, and saying one to another; Lift up your gates and receive the Mother of the Maker of heaven and earth. Let us praise with song her sanctified, noble body, which contained the Lord, invisible to us. Therefore we, too, celebrate your memory, O all-praised one, crying; Exalt the state of Christians and save our souls."