The term "Elessar" is elvish for "Elfstone;" it was given to Aragorn by Galadriel of Lothlórien. When parting from her land, the Lady gave each of the Company of the Ring a gift. To Aragorn she gave a silver brooch inset with a green stone, and with it his title.
When Aragorn took the throne of Gondor, he assumed the name King Elessar, and this is how he is referred to in most histories of Middle-Earth and Arda.