It was founded at the end of the Second Age (S.A. 3320) by Elendil, whose sons founded Gondor at the same time. The history of the two kingdoms is intertwined; both kingdoms are known as the Realms of the Dúnedain in Exile.
Before the foundation of Arnor there was already a sizable Númenórean population living there, a result of the slow emigration of Númenóreans which had started under Tar-Meneldur and Tar-Aldarion. Before the arrival of the Dúnedain Arnor was home to Middle Men of Edain stock, and the early colonists soon interbred with the indigenous population. Arnor was originally favoured over the more southern regions (Gondor) because the Elves under Gil-galad lived near it across the river Lhûn. However, in later days after the Númenóreans fell under Sauron's shadow they settled more to the south. This led to a situation where Elendil arrived in an area populated by people who, unlike his own Númenóreans, were mainly still friends with the Elves, and unlike Gondor to the south in Arnor much knowledge of the Elder Days was preserved.
Arnor's second king Isildur (also King of Gondor) was killed in T.A. 2 by Orcs in the disastrous Battle of the Gladden Fields. His three eldest sons were killed with him, but the fourth and youngest, Valandil, survived and became king of Arnor (T.A 2 or 10?).
After the death of its tenth king, Eärendur, in T.A. 861, Arnor was shaken by civil war between the three sons of Eärendur. The eldest son, Amlaith, claimed Kingship over all Arnor but was reduced to only ruling the region of Arthedain as his kingdom, while the other sons founded the kingdoms of Cardolan and Rhudaur.
Arnor was refounded in name by Arthedain's king Argeleb I, when Cardolan placed itself under the suzerainty of Arthedain. However, even Arthedain was eventually destroyed. The people of Arnor were mostly wiped out by the wars, but the Hobbits survived in the Shire, men survived in Bree and probably other villages, and the Dúnedain of Arnor created new homes in the Angle south of Rivendell, where some of them became known as the Rangers of the North.
Aragorn II as King Elessar refounded the Kingdom of Arnor as part of the Reunited Kingdom, and again made Annúminas his capital city. After the fall of Sauron Arnor was safe again for human population, and although it remained less populated than Gondor to the south in time Arnor became a more densely populated area again, even if it had dwindled in size due to the independence of the Shire.
Arnor was also the name of a software house which developed and published a range of business and utility software in the 1990s. Their Protext word processor sold over 100,000 copies on a range of platforms including the Amstrad PCW and MS-DOS.