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

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

Many of the systems had different names for the months:

Shire Calendar	Bree Calendar	Sindarin (Used in Gondor)	Westron (Quenya Names)
Afteryule	Frery	Narwain	Narvinye
Solmath	Somath	Ninui	Nenime
Rethe	Rethe	Gwaeron	Sulime
Astron	Chithing	Gwirith	Virisse
Thrimidge	-	Lothron	Lotesse
Forelithe	Lithe	Norui	Narie
Afterlithe	Mede	Cerveth	Cermie
Wedmath	-	Urui	Urime
Halimath	Harvestmath	Ivanneth	Yavannie
Winterfilth	Wintring	Narbeleth	Narquelie
Blotmath	Blooting	Hithui	Hisime
Foreyule	Yulemath	Girithron	Ringaire

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

Shire	(At the Time of the War of the Ring)	Sindarin	Quenya	Dunadan	Translation
Sterrendei	Sterday	Orgilion	Elenya	Elenya	Star-Day
Sunnedei	Sunday	Oranor	Anarya	Anarya	Sun-Day
Monendei	Monday	Orithil	Isilya	Isilya	Moon-Day
Trewesdei	Trewsday	Orgaladhad	Alduya	Aldea	Tree Day (Numenorean) or Trees-Day
Hevenesdei	Hevensday (or Hensday)	Ormenel	Menelya	Menelya	Heaven's-Day
Meresdei	Mersday	Orbelain (or Rodyn)	Valanya (or Tarion)	Valanya	Valar-Day
Highdei	Highday	Oraearon	Earenya	Earenya	Ship-Day (Added by the Numenoreans)

Our Days:	Shire Calendar:
Monday	Monday
Tuesday	Trewsday
Wednesday	Hevensday
Thursday	Mersday
Friday	Highday
Saturday	Sterday
Sunday	Sunday

Each of these calendars had a deficit of days, and each calendar dealt with it differently. The Numenorean 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.