The world knows it as New Year's Day. Our holy Mother Church knows it by many names.
First and foremost, it is the “Octave-day” or eighth day of Christmastide. Such was its name in the earliest liturgical books, thus remembered as the day of Circumcision, when a son of Israel was marked according to the Law. (It hurts just thinking about it.) In both forms of the Roman Rite, the brief account from Luke is proclaimed:
At that time, 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. (2:21)
In the reformed Missal, the day is primarily known as the Solemnity of Mary the Mother of God. While appearing as a break in tradition, it is a reminder of the Marian emphasis of the Feast, as found even in the orations of the pre-conciliar Missal. It was the tradition in Rome, that the Pope would go to one of the many churches in the city, whichever was the "Station" for that particular feast -- in the case of this one, the Basilica of Saint Mary Major.
But wait, there is so much more...
In the East, today is known not only for the Circumcision, but as the Feast of Saint Basil the Great, bishop of Caesarea in the fourth century, and one of the great Fathers of the Eastern Church. Today is when the Greeks would traditionally exchange gifts. For many years, when I couldn't meet with Paul for Christmas (and as he was raised in the Byzantine Rite of his mother), I would make an occasion of this day.
With all that arcane information, you still have to admit that four names for one day is a lot. And to think the year is just getting started.