The title of Mary as "Mother of God" was one of those puzzling peculiarities of Catholicism, as far as I was concerned back in my Baptist days. Jesus was God, and Mary was Jesus' mother, but how could she be mother of, well, all of God (who is infinite and timeless)? Catholic moral teaching had a reassuring solidity to it, even if not all its rules then struck me as necessary, but the Catholic way of thinking about the Virgin Mary I found downright disquieting.
Part of the answer is that Mary is in fact not mother of God the Father, nor of the Holy Spirit. But because Christ's divine nature, "eternally begotten of the Father," is united with his human nature in a single person (as solemnly affirmed by the fifth-century Council of Ephesus), his mother is truly the Mother of God.
The best short summary I've found of the doctrine is a General Audience by Pope John Paul II on 27 November 1996. And Jeff Mirus has a lovely compilation of reflections on Our Lady.
Suffice it to say a devotion to Mary, Mother of God, is an aspect of my faith I now treasure.
Holy Mary, Mother of God, pray for us sinners, now and at the hour of our death.