It was founded at the end of the Second Age (S.A. 3320) by Elendil, who 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.
Its 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.
See Kings of Arnor.