There were many calendars and systems for dating throughout the history of Middle-Earth, including:

Shire-Reckoning
Stewards Reckoning
Kings Reckoning
Númenorean Reckoning
New Reckoning (After the War of the Ring)

Many of the systems had different names for the months:

The Eldar had a 6 day week, which was changed by the Númenoreans into a seven day week. They also changed the names of some of the days.

Each of these calendars had a deficit of days, and each calendar dealt with it differently. The Númenorean Reckoning (or Kings Reckoning) dealt with it by adding a day every thousand years, but that left a small amount over, about 5 hours. This was corrected with the adaptation to the Steward's Reckoning. This one had such a tiny deficit, that by the War of the Ring it had not yet grown to one day. Each month had 30 days and the remaining days were holiday outside of the months.

The Shire Calendar used a similar system, save that the extra days were in two groups: Yule and Lithe.
Also, Year one in the Shire Reckoning was equivalent to 1601 of the Third Age in the Steward's Reckoning.

The New Reckoning
After the War of the Ring, the New Year began on March 25, the day that Sauron was destroyed. The month names remained the same, however they now began 5 days earlier than before.