In The Lord of the Rings: The Fellowship of the Ring why was Aragorn an exile at the beginning? Did a certain event cause him to become an exile?
Quoting from Wikipedia:
The son of Arathorn II and his wife Gilraen, Aragorn was born on 1 March, T.A. 2931.1 Through his ancestor Elendil (whom he closely resembled) Aragorn was a descendant of the first king of Númenor, Elros Tar-Minyatur; the twin brother of Elrond.
When Aragorn was two years old, his father was killed while pursuing orcs. Aragorn was afterwards fostered in Rivendell by Elrond. At the request of his mother, his lineage was kept secret, as she feared he would be killed like his father and grandfather if his true identity as Isildur's heir became known. Aragorn was renamed Estel ("hope" in Tolkien's invented language of Sindarin) to hide his existence from Sauron and his servants. He was not told about his heritage until he came of age in 2951.
Elrond revealed to Aragorn his true name and ancestry, and delivered to him the shards of Elendil's sword Narsil, and also the Ring of Barahir. He withheld the Sceptre of Annúminas from him until he "came of the right" to possess it. Aragorn met and fell in love with Arwen, Elrond's daughter (whom he mistook for Tinúviel), when she had returned from Lórien, her mother's homeland.
Aragorn thereafter assumed his role as the sixteenth Chieftain of the Dúnedain, the Rangers of the North, and went into the wild, living with the remnants of his people, whose kingdom had been destroyed through civil and regional wars centuries before.
I believe there is no need to explain more.
I can think of two different reasons why Aragorn was called an Exile:
A King without a Kingdom
Arnor eventually suffered from civil war and split up in smaller Kingdoms. All of the smaller Kingdoms were defeated by the Witch King. The Fall of Arnor left the House of Isildur without a Kingdom.
At the time of the War of the Ring, Aragorn was the heir of the House of Isildur. Because he was a King without a Kingdom, he may have been called an Exile.
An Exile from Númenor
The Kingdom of Númenor was an island raised by the One God as a reward for the faithful men who fought in the War against Morgoth. As years went on, the Kings of Númenor became ever more arrogant. In the end, they sent a large Armada against the Mayor Gods, hoping to wrest immortality from them.
The Mayor Gods called for help of the One God, and he destroyed the Armada and sunk the island of Númenor. Just before that, a few of the remaining faithful men had fled Númenor. Their leader was Elendil (the same as above.)
Because Aragorn was a descendant of Elendil, he could be said to be a Númenorean in Exile.