"You are Isildurs heir, Not Isildur himself." Arwen to Aragorn.

Aragorn (son of Elenthil, heir to Isildur) is the last remaining heir to the throne of Gondor.
He has however he has given up the throne and gone to live in the wild because of his fear of his connection to the one ring.
Aragorn is connected to the ring because of the pledge made by Isildur after the finding of the one ring after the beating of Sauron at the end of the second age. Isildur was taken by Elrond into the heart of Mount Doom where he was told to cast the ring into the fires thus destroying Evil forever. Isildur however did not destroy the ring and pledged that all who follow in his bloodline would be drawn to it. The ring however betrayed Isildur to his death and the ring was lost until it was found by its new owner.
Because of this Aragorn has left Gondor to live in the wild as a Ranger. However will he ever return to Gondor and take up his throne.
After finding the four Hobbits in Bree, Aragorn took them to the Elvish settling of Rivendell where he also attended the secret council meeting to where representitives of all middle Earth must decide how to deal with the threat of Mordor and what to do with the one ring.