"'Behold the handmaid of the Lord: Be it done unto me according to Thy word.'

And the Word was made Flesh: And dwelt among us."

463 Belief in the true Incarnation of the Son of God is the distinctive sign of Christian faith: "By this you know the Spirit of God: every spirit which confesses that Jesus Christ has come in the flesh is of God."

483 The Incarnation is therefore the mystery of the wonderful union of the divine and human natures in the one person of the Word.

745 The Son of God was consecrated as Christ (Messiah) by the anointing of the Holy Spirit at his Incarnation (cf. Ps 2:6-7).

607 The desire to embrace his Father's plan of redeeming love inspired Jesus' whole life, for his redemptive passion was the very reason for his Incarnation.

It simply means that "God became Man".  

But what it implies is far from simple.  In fact, there is no greater miracle that we believe in the Catholic Church, than that God became "one of us" through the power of the Holy Spirit and the cooperation of the Blessed Virgin.  It precedes in importance even the Holy Eucharist, for we couldn't receive His Body and Blood if He had not first assumed the flesh we would consume:   

"we do see Jesus 'crowned with glory and honor' because he suffered death, he who 'for a little while' was made 'lower than the angels,' that by the grace of God he might taste death for everyone."  (Hebrews 2:9)

When Did the Incarnation Take Place?

The Incarnation of Our Lord is depicted in every representation of the mystery of the Annuciation to Mary.  The Incarnation did *not* take place at the Nativity.  It took place when Jesus was conceived in the womb of the Blessed Virgin Mary by the Holy Spirit.

If We Celebrate the Conception of God in the Flesh, Aren't We Affirming That Life Begins at Conception?

It is impossible that the Incarnate God, even in embryonic stage, did not exist. His life in the flesh began at His conception in the womb of Mary.  So too, our fleshly lives begin at conception, and therefore, every conception enjoys the dignity of being made in the Image of God, at every stage.  Each human being has an identity in the Creator, and cannot be reduced to mere "tissue" or "cells".  To do so rejects God's own Incarnation at the Annunciation to Mary.