There have been several year-numbering systems in use over the course of the centuries. In my own area, the P.D. dating system -- that is to say, the chronographic system used by the Dwemer -- commenced about 4500 years ago, ca. 700-600 ME (plus or minus a century or so).
Aside from that there have been four separate dating systems in widespread use -- consecutively, not concurrently. Each subsequent calendar era was pronounced upon the occasion of the beginning or end of one political dynasty or other. Thus the First Era began with the declaration of the Camoran Dynasty; the Second Era began a little under 3000 years later with the end of the Reman Dynasty and beginning of the period of the Akaviri Potentates; and the Third Era began 896 years after that, with the founding of the Septim Dynasty, which lasted 433 years. The present calendar era dates back 201 years, beginning with the end of the Septim Dynasty and the interregnum that followed after the Oblivion Crisis. There has never been any serious dispute over the use of the Second to Fourth Era systems, at least.
Having said that, naturally there are competing systems in play. For some purposes the Aldmeri Dominion has reverted to the Camoran (First Era) calendar era, even though the Dominion has only been in existence since shortly after the beginning of the Fourth Era, but for diplomatic and trade purposes the Fourth Era system retains its dominance pretty much everywhere. And there are still some separatist guerillas in Pellitine who use the Elsweyr Confederacy dating system, which dates back to 2E 309; but that's only for domestic purposes.
The current dating system is based on major events and wars. The first known dating system was put in place several centuries after the Breaking of the World. The Breaking was such an apocalyptic event that it was clear to everyone that the Age of Legends had passed and a new age had begun.
The first calendar system to be adopted in this Age, the Toman Calendar, counted years "After Breaking" (AB). Since everything was lost in the Breaking, the year 1 AB was arbitrarily assigned to some 200 years before the adoption of the calendar. We're not exactly sure on the exact year this happened, as the Trolloc Wars destroyed all records available.
A new calendar, devised by Tiam of Gazar, started counting years from the end of the Trolloc Wars to celebrate the freedom from the trolloc threat. Years in this calendar were called "Free Year" (FY). The Gazarian Calendar was widely adopted within 20 years after the end of the war.
The calendar currently in use, the Farede Calendar, was created by a Sea Folk scholar named Urin din Jubai Soaring Gull. This calendar started at the end of the War of the Hundred Years and numbered years in the "New Era" (NE). The calendar is named after the first Panarch of Tarabon, Farede. From the important trading and intellectual center of Tanchico the use of this calendar spread across the world, gaining widespread adoption within 50 years.
Arthur Hawkwing attempted to create his own calendar based on the founding of his empire. Years were named "From the Founding" (FF). This calendar failed to gain widespread adoption and is now only used by scholars.
As I said above, due to the huge wars and large scale destruction between the different calendars, it is hard to say which year is which in each calendar. To give a rough approximation though:
The Shire-Reckoning is the main calendar dating system used in the primary sources that I study, with Year 1 of the Shire-Reckoning being the year that Hobbits first settled the Shire under the direction of Marcho and Blanco, following the grant of the Shire to them by King Argeleb II of Arthedain. This was 1420 years prior to the Ringbearer taking Isildur's Bane to Orodruin.
However, you bring up a fair point- what year would it have been to Marcho and Blanco, and what year would it have been to the men and hobbits of Bree before the settlement of the Shire? It is likeliest, since this is the most common calendar among the Adunaic speaking peoples, that they would have been using the King's Reckoning in Bree, a Calendar first devised according to my estimations upon the settlement of the island of Numenor. This dates the year according to its relationship to the beginning of an Age of the world. The beginning of the Second Age is traditionally considered to be the banishment of Morgoth to the Void following the conclusion of the War of Wrath, though some less traditional scholars have instead proposed that it 'really' began with the settlement of Numenor and the ethnogenesis of the Adunaim. Thus, for events between this date and the first defeat of Sauron by the Last Alliance, the King's Reckoning dates by 'the Xth year of the Second Age' or ' S.A X' (Second Age X). Then, between this date and the passing of the Ringbearers, Mithrandir, the Lady Galadriel, and Master Elrond into the West, we have the years of the Third Age. It is in the Third Age, according to this reckoning, that Marcho and Blanco would have lived in Bree, and it would have been the 1601st year of the Third Age when they settled the Shire, or T.A 1601. This was likely the year to them when the Shire-Reckoning era was created.
The 'War of a Thousand Suns' is normally taken as the terminus for the Old World calender, and while the old dating system still persisted in many parts of the Plainfolk, the George Washington Jefferson the 1st's death is normally considered to be the beginning of the new calendar for most of the Inner and Outer Home states, and some of the New territories (although Colorado is so heavily disputed, I suspect that there will be no resolution to this).
I've heard that the Da-Tsuni Shogunate use a different a different dating system than either of the above, although I've only read several Ne-Issan monographs on the issue, and there isn't an academic consensus on the matter, unless Amexico has something hidden in the archives (due to be released somewhere in 3530).