Ancient Romans used to start their new year in March, since their first calendar only had 10 months which ended in December, approximately harvest time, so people can celebrate at the end of the year with a good harvest. Then, two months of winter will pass by and they start a new year.
This configuration of the early Roman calendar had most likely been adopted from other dating systems like those of the Greeks, Etruscans, and Babylonians. The Roman simply named their months after gods and goddesses like the war god Mars for March, and Juno for June. The rest of the months had been named after the numbers such as septem for seven, and so on.
Later on, this 10-month calendar was replaced by a lunar calendar which added two new months, each consisting of 28 days: Ianuarius (named after the god Janus) and Februarius (a reference to the Roman purification festival, Februa).
The problem with this new calendar, they soon found out, was that it did not sync with the seasons. They tried to rectify this by adding an extra month called Mercedonius, every two or three years, which just made the matter more confusing.
It wasn't until Julius Caesar had commissioned a new standard calendar in 45 BC. Designed by Sosigenes, the Julian calendar consisted of 365 days with a leap year every four years. This was more consistent with the seasons, and finally synced with the sun.
January 1st became the beginning of the new year as well, as opposed to the original March. It was in celebration of the day that consuls, the highest elected public official in the Roman Republic, took office.
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