The title of Mary as Mother of God proclaims clearly that the One to whom she gave birth truly is God. In fact, He is one Divine person who is both fully God and fully Man. The feast day is a clear example of how devotion to Mary actually helps us to know and love her Son better. True devotion to Mary never takes away from our relationship with Jesus Christ, but protects and even fosters it.

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Resources from the Catechism
On Nestorianism (click here)
On Mary as the Theotokos (Mother of God) (click here)

From the Catholic Encyclopedia entry for the "Blessed Virgin Mary"
Mary's Divine motherhood is based on the teaching of the Gospels, on the writings of the Fathers, and on the express definition of the Church. St. Matthew (1:25) testifies that Mary "brought forth her first-born son" and that He was called Jesus. According to St. John (1:15) Jesus is the Word made flesh, the Word Who assumed human nature in the womb of Mary. As Mary was truly the mother of Jesus, and as Jesus was truly God from the first moment of His conception, Mary is truly the mother of God. Even the earliest Fathers did not hesitate to draw this conclusion as may be seen in the writings of St. Ignatius, St. Irenaeus, and Tertullian. The contention of Nestorius denying to Mary the title "Mother of God" was followed by the teaching of the Council of Ephesus proclaiming Mary to be Theotokos in the true sense of the word.