In the Old Law, by the rite of Circumcision, every male Jew became a member and shared in the privileges and blessings of the chosen people of God. A Jew who failed to be initiated by the ceremony was excluded. Our Lord was Son of God by nature, and absolutely sinless, and therefore did not need adoption into the membership of God's children. Yet, He submitted to the law. The Church also honors on this day the holy name of Jesus, given to the Divine Child at the Circumcision.
Today is especially important for two reasons: on the eighth day the Christ Child received the Holy Name of Jesus, and on the eighth day He first shed His Precious Blood for us, of which the Church teaches that one drop would have sufficed for our salvation. For this reason, the whole month of January is dedicated to the Holy Name and Childhood of Jesus (tomorrow is the Feast of the Holy Name). Why the Novus Ordo Calendar moved the Feast of the Motherhood of Mary from Oct 11 to Jan 1 is a mystery to me; I wonder if anyone has seen/heard a justification offered for this move? Practically speaking, it would be hard to dispute that this move has led to a neglect of the salvific significance of the Circumcision and the Holy Name.
I highly recommend the wonderful source of traditional Catholic information that is Fish Eaters for their page on the Feast of the Circumcision, and on the practice of (or avoidance of) circumcision.
And after eight days were accomplished, that the child should be circumcised, his name was called Jesus, which was called by the angel, before he was conceived in the womb (Luke 2:21).