Eldarion (Quenya "Son of Elves") was the only recorded son of Arwen and Aragorn.
Eldarion's birth was significant in that he represented the union of the lines of Elros (through Aragorn) and Elrond (through Arwen) separated for millennia.
He was born in the Fourth Age, and became King of the Reunited Kingdom of Gondor and Arnor after his father died. He was also recorded to have several sisters.
105 years after the fall of the Dark Tower, he encountered a renewal of Morgoth-worship during his reign.
His name in Quenya means 'Scion of Eldar'. In Letter 338, J.R.R. Tolkien spelled this name Eldaron, this is probably a typographic error.
Portrayal in adaptations
In Peter Jackson's The Return of the King, Arwen is on her way to the Grey Havens to leave Middle-earth and Aragorn when she has a vision of her future son (not named in the film), which persuades her to stay. The boy is played in the film by Sadwyn Brophy.