What does it mean that the Virgin was assumed "in body" in Heaven?

The Catholic Church has always taught, and still teaches that Mary, Mother God (in Greek: Theotokos) was assumed into Heaven completely. Her physical body was taken to Heaven, along with her immaculate soul.

We see this forshadowed in scripture, especially in the Old Teastament.

We read in Genesis 5:24, and Hebrews 11:5 that Enoch was bodily assumed into heaven without dying.

Enoch walked faithfully with God; then he was no more, because God took him away. (Gen. 5:24)

By faith Enoch was taken from this life, so that he did not experience death: “He could not be found, because God had taken him away.” [a] For before he was taken, he was commended as one who pleased God. (Heb. 11:5)

Would God do any less for Mary the Ark of the New Covenant?

2 Kings 2:11-12 tells us that Elijah was assumed into heaven in a fiery chariot. Jesus would not do any less for His Blessed Mother.

And it came to pass, as they still went on, and talked, that, behold, there appeared a chariot of fire, and horses of fire, and parted them both asunder; and Elijah went up by a whirlwind into heaven. And Elisha saw it, and he cried, My father, my father, the chariot of Israel, and the hors*m*n thereof. And he saw him no more: and he took hold of his own clothes, and rent them in two pieces. (2 Kings 2:11- 12)

Mary's Assumption has been explictly taught by the Church Fathers from very early on in the Church's history.

St Gregory of Tours writes in the 6th century:

"[T]he Apostles took up her body on a bier and placed it in a tomb; and they guarded it, expecting the Lord to come. And behold, again the Lord stood by them; and the holy body having been received, He commanded that it be taken in a cloud into paradise: where now, rejoined to the soul, [Mary] rejoices with the Lord's chosen ones..." Gregory of Tours, Eight Books of Miracles, 1:4 (inter A.D. 575-593).

Two centuries later, John Damascene writes:

"It was fitting that the she, who had kept her virginity intact in childbirth, should keep her own body free from all corruption even after death. It was fitting that she, who had carried the Creator as a child at her breast, should dwell in the divine tabernacles. It was fitting that the spouse, whom the Father had taken to himself, should live in the divine mansions. It was fitting that she, who had seen her Son upon the cross and who had thereby received into her heart the sword of sorrow which she had escaped when giving birth to him, should look upon him as he sits with the Father, It was fitting that God's Mother should possess what belongs to her Son, and that she should be honored by every creature as the Mother and as the handmaid of God." John of Damascene, Dormition of Mary (PG 96,741), (ante A.D. 749).

There are many other writings by Church Fathers attesting to the Tradition of the Assumption...too many to list here.

Perhaps you can see how this dogma was believed and taught by the Church well before the recent precisions you have cited.

Yes, this refers to Mary's material, human body. The Catholic Church teaches that Mary was assumed into heaven, body and soul.

For an explanation of the reasons behind this teaching, you would do well to read the decree "Munificentissimus Deus" of Pope Pius XII, in which he officially defined this dogma of the assumption. The "punch line" is "by the authority of our Lord Jesus Christ, of the Blessed Apostles Peter and Paul, and by our own authority, we pronounce, declare, and define it to be a divinely revealed dogma: that the Immaculate Mother of God, the ever Virgin Mary, having completed the course of her earthly life, was assumed body and soul into heavenly glory." This is from section 44 of the decree, so you can see that a lot of explanation preceded it.

The whole decree is available on the Vatican web site, at http://www.vatican.va/holy_father/pius_xii/apost_constitutions/documents/hf_p-xii_apc_19501101_munificentissimus-deus_en.html (in English). I should probably mention explicitly that, although this decree dates from 1950, the Church celebrated the feast of the assumption for many centuries before that. As far as I know, the Eastern Orthodox churches agree with this teaching, but protestants do not.