The Third Age (III) is the time period in which the Lord of the Rings (and LotRO by extension) takes place, when the Fellowship of the Ring go forth on their journey to destroy the One Ring and save the world from Sauron.
The Third Age begins with the end of the siege of Barad-dûr and the temporary defeat of Sauron. Isildur takes the One Ring, sets affairs in order in Gondor, and in 2 T.A. he is killed while returning north in the Battle of the Gladden Fields, where the One Ring was lost in the Anduin and passed out of knowledge for many years. The Third Age covers 3021 full years and ends with the Ring-bearers departing to the Grey Havens. — 
Arnor & Gondor
Though Isildur and his eldest sons never returned to the north, his youngest son Valandil eventually took up the rule of Arnor in Annúminas. Arnor endured until 861, when King Eärendur died and the kingdom was divided between his three sons into the three smaller realms of Arthedain (which included much of what is now the North Downs and Evendim), Cardolan (south of the Lone-lands), and Rhudaur (around the Trollshaws, sharing a border with Angmar).
The royal line soon failed in Rhudaur and Cardolan and eventually the kings of Arthedain claimed lordship of all of old Arnor. Cardolan was amenable to alliance at the very least, but Rhudaur refused and joined forces with Angmar to make war against Arthedain and Cardolan. The Witch-king of Angmar invaded Arnor in 1409, but the northern Dúnadan-kingdoms held out until their last holdouts were finally overrun in 1974. Though the Witch-king was defeated at the Battle of Fornost the next year, the Northern Kingdom was ended and the Dúnedain went into hiding, preserving the line of Isildur in the Chieftains of the Dúnedain beginning with Aranarth.
After the War of the Last Alliance, Gondor was ruled by Meneldil, son of Anárion. Gondor rose to great heights, becoming a great naval and military power in the south. In 1432 the Kin-strife began, a civil war that for a time drove Eldacar Vinitharya, son and heir of King Valacar and Vidumavi, princess of Rhovanion, from Gondor. Eldacar returned and retook his throne in 1447 from Castamir the Usurper. The wounds from this have still not entirely healed.
In 1940, Fíriel, daughter of King Ondoher, wed Arvedui, King of Arthedain. When her father and brothers, Artamir and Faramir, were killed in battle with the Wain-riders, Fíriel and Arvedui pressed Fíriel's claim to the throne of Gondor by the laws of Númenor, but the council of Minas Tirith elected instead to crown Eärnil II, a more distant relation but still of the royal house and a proven general. Eärnil II's son, Eärnur, also distinguished himself in battle, including riding against the forces of the Witch-king at Fornost in 1975. Eärnur was issued a challenge by the Witch-king, and in 2050 he rode to Minas Morgul in answer and was lost. Mardil began the line of the Ruling Stewards, which continued to rule Gondor until the end of the Age, when Faramir, son of Denethor, ceded the crown to Aragorn.
The kingdom of Rohan as it is known today began in 2510, when Cirion, Steward of Gondor, gifted the near-empty province of Calenardhon to Eorl the Young and his people for their aid on the Field of Celebrant. A history of the Rohirrim is told in the tapestries of Meduseld- of the building of the Golden Hall, their many conflicts with the Dunlendings (many of whom the Eorlingas drove beyond the River Isen while establishing Rohan), Saruman moving into Orthanc, and the story of Folca.
Before the Battle of Celebrant, the Éothéod lived well to the north, far up the river Anduin. Around 1900 Marwhini of the Ai-thúda established Avabárg in the Vales of Anduin. By 1977, Frumgar led the Ai-thúda farther north, nearer to Mount Gundabad which had been weakened significantly by the fall of Angmar in 1975. Framsburg was established in the Wells of Langflood. Fram, son of Frumgar, had both friendship and enmity with the Dwarves of the region, arguing over the hoard of Scatha the Worm.
See also: Léothred
Though many of the Hól-budlan fought in the Last Alliance, there are few records of the Hobbits. Around 1000, the Harfoots were first noted in Eriador. A century or so later, a group of Stôrs settled in the Angle of Mitheithel and Dunland, at villages like Glynafon or Maur Tulhau. The Fallohides crossed the Misty Mountains around the same time, though some lingered at in the Wells of Langflood- Lyndelby still stands today. As the power of Angmar grew around 1300, many of the Hobbits migrated to Bree, and later in 1601 beyond the Baranduin to land granted them by Argeleb of Arthedain. This became the Shire.
When Angmar was near its height between 1300 and 1975, its power extended into Câr Bronach north of Mount Gundabad. The Zhélruka fought many battles against the Angmarim, breaching the pass of Rathad Caul and shattering the tower of Tûr Fúar, the Drearspire.
The Longbeard Dwarves had a largely uneventful first half of the Age, until 1980 when a Balrog appeared in Khazad-dûm and killed Durin VI. Thráin I, his grandson, established the kingdom under Erebor, though he later removed to the Grey Mountains, where more of his people had gone. The Longbeards remained there until 2589, when Vethúg Wintermind descended on Thikil-gundu and again drove the Longbeards from their home. They returned to Erebor until 2770 when Smaug the Golden ousted them from the Kingdom Under the Mountain. The Longbeards wandered for a time, establishing halls at Zudrugund and Thorin's Hall. Thráin II mustered an army from the surviving Longbeards and dwarves of the other six Houses to seek vengeance for his father, who was killed in Moria. The war was successful, if costly, and many orcs fled the Misty Mountains (and began to trouble Rohan) and finally ended with the Battle of Azanulbizar in 2799. It was still not until 2941 that Erebor was reclaimed and the neighboring human kingdom of Dale reestablished.
War of the Ring
The end of the Third Age is marked by the end of the Rings of Power- the destruction of the One Ring and Sauron as is detailed in The Lord of the Rings, the waning of the power of the remaining Rings, and the eventual departure of the Ring-bearers for the Undying Lands.
Note on Timelines
With the exception of a handful of flashback quests and regions (Mordor Besieged, the High Elf introduction, Azanulbizar, T.A. 2799, etc.), the story of LotRO largely takes place during 3018 and 3019 T.A., paralleling and at times touching on the story of the Fellowship of the Ring. Volumes I-V of the Epic follow the books more closely, while the Black Book of Mordor deals with the immediate aftermath of the War of the Ring and the Legacy of Durin and the Trials of the Dwarves deals largely with the reclamation of Gundabad, both of these well outside the scope of the books. This is not to say we've reached the end of The Return of the King just yet- Volume V begins with the Midsummer wedding of Aragorn and Arwen (chronologically before Chapter 13 of the Black Book), and there is much still to come.