While we know precious little of his life, we do know from the Liber pontificalis that Saint Sylvester was the son of Rufinus a Roman. When Pope Miltaides died, Saint Sylvester was elected Bishop (having been a Presbyter, or in the early Church an officer/minister in between a Bishop and a Deacon) and was consecrated on January 31, 314 AD as the Bishop of Rome. Although he was invited to the Council of Nicea, he did not attend but rather sent two legates (Vitus and Vincentius) and gave his full approval over the Council's decrees and the Nicene Creed. His time as Pope coincided with the reign of Constantine and the approval of Christianity in the Roman world.
Saint Sylvester also presided over the writings of the first martyrology (the recording of the Roman Martyrs). He built many churches over the graves of Martyrs to honor them. We also know that Saint Pope Sylvester oversaw the building of the Basilica of Saint John Lateran, the Basilica of the Holy Cross in Jerusalem, and Old Saint Peter's Basilica during his pontificate.
Several legendary stories record miracles of Saint Sylvester. In one, the Emperor Constantine was cured of leprosy when he was baptized in Holy Water administrated by Saint Sylvester. In the Emperor's gratitude he confirmed that the Bishop of Rome was the primate above all Bishops, resigned his Imperial insignia, and walked before Saint Sylvester's horse holding the Pope's bridle as the papal groom. In return, Saint Sylvester offered the crown back to Constantine who began to live there in Constantinople, placing a doctrine that the Pope is supreme above all earthly rulers including the Roman Emperor.
Today, Saint Sylvester's day is celebrated as the Feast of Saint Sylvester I on December 31st, the day of his burial in the Catacomb of Priscilla (though his relics were later transferred by Saint Pope Paul I to the basilica of San Silvestro in Capite). Liturgically, this also falls on the 7th day of Christmastide in the Western Roman Catholic church (the feast day falls on Jan 2 for the Eastern churches). Rather than calling it New Year's Eve, many partakers of this feast day refer to the day as Sylvester's Day.
In Vienna, it is a famous practice to walk pigs on leashes around the town for the hope of good luck in the coming new year. Many Christian's also practice the custom of Bleigissen using Silvesterblei (Silvester Lead) in which the lead is melted over a flame in an old spoon before being dropped into a bowl of cold water. The shape of the cooled lead determines the fortune for the next year - if it forms a ball, luck is coming (literally rolling one's way) and a start means happiness.
In Belgium, a maiden who does not finish her work by sunset on Saint Sylvester's day won't be married in the coming year. In Brazil there are fireworks (similar to New Year's Eve in America). Practitioners in Italy often eat lentils and slices of sausage (because they look like coins symbolizing good fortune for the coming year).
The Swiss have a tradition where the children of the family compete to see who can wake up earliest - the one who wakes up the latest is playfully jeered throughout the day. There is also a famous custom in Urnasch, Switzerland called Silvesterklausen where men and boys disguise themselves in costumes (called Silvesterchlaus), wear harnesses of heavy bells and walk in groups from house to house singing different yodels. When they arrive to a house, they ring the bells and the family comes out with drinks. When the yodels are done, each is offered a drink and a handshake before marching off to the next house.