The Roman calendar that Europe inherited began as a ten month year opening in March, with the span after December left as an unnumbered winter gap. Roman tradition credited the king Numa Pompilius with adding two months to close that gap, and the first of them, Ianuarius, took its name from Janus, the god of doorways, arches and transitions, portrayed with two faces so that he looks into the year behind and the year ahead at once. Ovid opens the Fasti, his verse calendar of the Roman festivals, by asking Janus himself to explain the month, and has the god answer that every beginning belongs to him.
January did not immediately open the civil year everywhere. Roman consuls entered office on the first of January from 153 BCE, which fixed the date as the practical new year of the Roman state, but medieval Christian kingdoms variously began the year at Christmas, at the Annunciation on the twenty fifth of March, or at Easter. England and its colonies only moved New Year's Day back to the first of January in 1752, when they adopted the Gregorian reform, which is why English documents from earlier centuries can carry two year numbers between January and March.
The month's Roman character was largely civic and domestic: the Agonalia of the ninth of January honoured Janus, and the Carmentalia of the eleventh and fifteenth honoured the birth goddess Carmenta. The Kalends of January, when Romans exchanged sweet gifts and well wishes for luck in the opening year, is a recognisable ancestor of the secular New Year customs still kept across the world.
Sources FastiOvid with The Oxford Companion to the Year: An Exploration of Calendar Customs and Time-ReckoningBonnie Blackburn and Leofranc Holford-Strevens