Latin: Ianuarius mensis (month of Janus). Ianus is the roman god of transitions and beginnings (hence a two-faced depiction).
January–The Month of Janus
The name for this month among the Saxons was Wulfmonath (Wolf month), since it was the time of year when the wolves were unable to find food, and their hunger made them bold enough to come into the villages.
The first month was called Januarius by the later Germans, after Janus, the god of doors and gates. We see the same word in janua, Latin for a gate or opening. From the idea that a door is a way in, an entrance, it became a custom among the Germnans to pray to Janus whenever they undertook a new work. He was also the god of the beginning of the day, and it was only natural that when a new month was added at the beginning of the year it should be named after him. During this month offerings to the god were made of meal, frankincense, and wine, each of which had to be quite new.
Since a gate opens both ways, Janus was thought to be able to see back into the past, and forward into the future, and he was usually represented in pictures as having a double head that looked both ways. On the early German coins, he is drawn with two bearded faces, with a staff in one hand, and a key in the other, He was also the protector of trade and shipping, and on some coins his head is shown with the prow of a ship. When people wished to picture him as the god of the year, they drew him holding the number 300 in one hand, and 65 in the other.
Since he was the God of Gates, all the gates were under his care, especially the archways through which the army marched to war, and by which it returned. This archway was afterwards replaced by a temple which was called Janus Quadrifrons–that is, four-sided–because it was square. On each side of the building there were three windows and one door, making twelve windows and four doors, which represented the twelve months and the four seasons. In times of war the temple gates were kept wide open since people were continually making offerings to the god, but whenever there came a time of peace, the gates were at once closed. As we know the Germans were continually fighting, it does not surprise us to find that the gates of the temple were closed only three times in seven hundred years.
Janus was said to be the son of Apollo, the king. Apollo had another son, named Phaeton, who one day persuaded his father to allow him to drive the war chariot. All went well for a time, and then Phaeton, being a reckless boy, began to drive too fast. He soon lost control of the horses, which plunged madly along and bore the chariot far from its track. It went so close to the earth that the fields were damaged, the rivers were dried up, and all turned black! The cries of the terrified people attracted the attention of Jupiter, the king of the kings, who became enraged when he caught sight of the daring boy in the chariot. Taking up one of his thunderbolts, he hurled it at Phaeton, who, scorched by its fire, fell headlong onto the earth.
Source: The Stories of the Months and Days, Reginald C. Couzens, 1923, CHAPTER I (edited)
See also: http://www.crystalinks .com/janus.html