The Roman god who gave January its name and was honored at the year's beginning was Janus.
Janus was associated with doors, gates, transitions, beginnings, and endings. He was commonly depicted with two faces, allowing him to look toward the past and the future. That imagery made him especially appropriate for the passage from one year into the next.
The month January comes from the Latin Ianuarius, meaning the month of Janus. Roman religious observances at the start of the year included prayers and offerings to him, and the name survived as calendars and month names developed through later European traditions.
Janus is sometimes confused with Saturn because the Roman festival Saturnalia occurred in December, near the modern New Year. They were different gods with different roles. The modern New Year's Eve party is not a direct survival of one single Roman ceremony, but its language of transition strongly echoes Janus's traditional symbolism.