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From 'Numa Pompilius' By Plutarch

According to legend, Numa Pompilius was the second king of Rome, succeeding Romulus. He is supposed to have reformed the calendar, and died died in 673 BC. Plutarch (46-127 AD) is describing his reign, and the calendar reforms.
This translation is by John Dryden.

He [Numa Pompilius] attempted, also, the formation of a calendar, not with absolute exactness, yet not without some scientific knowledge. During the reign of Romulus, they had let their months run on without any certain or equal term; some of them contained twenty days, others thirty-five, others more; they had no sort of knowledge of the inequality in the motions of the sun and moon; they only kept to the one rule that the whole course of the year contained three hundred and sixty days.

Numa, calculating the difference between the lunar and the solar year at eleven days, for that the moon completed her anniversary course in three hundred and fifty-four days, and the sun in three hundred and sixty-five, to remedy this incongruity doubled the eleven days, and every other year added an intercalary month, to follow February, consisting of twenty-two days, and called by the Romans the month Mercedinus. This amendment, however, itself, in course of time, came to need other amendments.

He also altered the order of the months; for March, which was reckoned the first he put into the third place; and January, which was the eleventh, he made the first; and February, which was the twelfth and last, the second.

Many will have it, that it was Numa, also, who added the two months of January and February; for in the beginning they had had a year of ten months; ... That the Romans, at first, comprehended the whole year within ten, and not twelve months, plainly appears by the name of the last, December, meaning the tenth month; and that March was the first is likewise evident, for the fifth month after it was called Quintilis, and the sixth Sextilis, and so the rest; whereas, if January and February had, in this account, preceded March, Quintilis would have been fifth in name and seventh in reckoning.

It was also natural that March, dedicated to Mars, should be Romulus's first and April, named from Venus, or Aphrodite, his second month; in it they sacrifice to Venus, and the women bathe on the calends, or first day of it, with myrtle garlands on their heads. But others, because of its being p and not ph, will not allow of the derivation of this word from Aphrodite, but say it is called April from aperio, Latin for to open, because that this month is high spring, and opens and discloses the buds and flowers.

The next is called May, from Maia, the mother of Mercury, to whom it is sacred; then June follows, so called from Juno; some, however, derive them from the two ages, old and young, majores, being their name for older, and juniores for younger men.

To the other months they gave denominations according to their order; so the fifth was called Quintilis, Sextilis the sixth, and the rest, September, October, November, and December.

Afterwards Quintilis received the name of Julius, from Caesar, who defeated Pompey; as also Sextilis that of Augustus, from the second Caesar, who had that title. Domitian, also, in imitation, gave the two other following months his own names, of Germanicus and Domitianus; but, on his being slain, they recovered their ancient denominations of September and October. The two last are the only ones that have kept their names throughout without any alteration.

Of the months which were added or transposed in their order by Numa, February comes from februa; and is as much a Purification month; in it they make offerings to the dead, and celebrate the Lupercalia, which, in most points, resembles a purification.

January was also called from janus, and precedence given to it by Numa before March, which was dedicated to the god Mars; because, as I conceive, he wished to take every opportunity of intimating that the arts and studies of peace are to be preferred before those of war. For this Janus, whether in remote antiquity he were a demigod or a king, was certainly a great lover of civil and social unity, and one who reclaimed men from brutal and savage living; for which reason they figure him with two faces, to represent the two states and conditions out of the one of which he brought mankind, to lead them into the other. His temple at Rome has two gates, which they call the gates of war, because they stand open in the time of war, and shut in the times of peace; of which latter there was very seldom an example, for, as the Roman empire was enlarged and extended, it was so encompassed with barbarous nations and enemies to be resisted, that it was seldom or never at peace. Only in the time of Augustus Caesar, after he had overcome Antony, this temple was shut; as likewise once before, when Marcus Atilius and Titus Manlius were consuls; but then it was not long before, wars breaking out, the gates were again opened.

Click here for a short description of where our month names came from.