Elladan and Elrohir were born to Elrond Halfelven and his wife, Celebrían, in the year TA 130. The twins were so much alike, both dark-haired, grey-eyed and elven-fair, that few could tell them apart. Indeed they seldom were apart. Never is there an instance where the brothers are not mentioned together in all of Tolkien's work.

In TA 2509, Celebrían, mother also to Arwen Undómiel, was travelling to Lórien when she and her party were assaulted by orcs in the Redhorn Pass. Celebrían was seized and later tormented by her captors. Elladan and Elrohir pursued the foul creatures and succeeded in rescuing their mother. She was taken back to Imladris where she was healed in body by Elrond, but lost all desire to remain in Middle Earth. The next year, she departed into the West. After this event, the brothers' war with the orcs never ceased.

Both brothers were also present in TA 2933 when Arathorn II, father to Aragorn II, later to be King Elessar Telcontar first King of the Reunited Kingdom, was slain by an orc-arrow that pierced his eye. Aragorn was but a child of two at the time and was after taken as a foster-son by Elrond. During his youth in Imladris, Aragorn rode often with the Sons of Elrond, learning much from them about hunting, war and the ways of the wild.

Elladan and Elrohir were with the Dúnedain when they rode to meet Aragorn in Rohan during the War of the Ring in TA 3019. The two were members of the Grey Company that took the Paths of the Dead in an attempt to cut off the Corsairs before they could assail Minas Tirith from the South. Elladan it was who took up the rear guard as the Company passed underground to gather the Army of the Dead. Elrohir bore to Aragorn the silver horn used to summon the Army of the Dead to the Stone of Erech.

A few years after the War, in TA 3021, Elrond passed into the West. With his passing, the Third Age came to an end and his children were given a choice. As long as Elrond remained in Middle Earth, his children had the life of the Eldar. But when Elrond departed, they must also depart or become mortal and die. Arwen chose to remain in Middle Earth for the love of King Elessar. The brothers delayed their choice for a while and dwelt in Rivendell for several years after the departure of their father. No one really knows what became of the brothers. Did they sail into the west along with their grandfather Celeborn or did they remain in Middle Earth as guardians of Rivendell untill they died? No one really knows. Let's hope it was not the latter.

More ponderings about the staying or leaving of the twins:

There was no reason for E and E (as we like to call them) to make the same decision as their sister Arwen. They just decided to stay in Middle Earth a bit longer and that is no reason to think that they would not sail into the west. I believe that the brethren sailed into the West at about the same time as their grandfather, Celeborn. After all, he came to live with them when Galadriel had already sailed.

Why did they chose to stay behind for a while?

Perhaps they still relished the fight against evil creatures and stuck around, knowing there would be a good deal of clean-up work to do. Or maybe they were not tired of the other races yet, being in the company of Elves, Men, Hobbits, Dwarves... All the things of Middle Earth that the Blessed Realm just did not have. But there really is no way of telling. I guess they both had their whims.

Extra:

Elladan means Elf Man, or literally Elf-Numenorean in the common tongue and Elrohir means Elf knight.

Pronounciation of the names:

Elladan: E'lladan and Elrohir: E'lroheer

Unfortunately for us they did not appear in the movie, and if they will appear in TTT and ROTK is something we hope and pray for and we can only wait.

They help the ancestors of the Rohan fight a battle against the orcs 500 years before the LOTR timeline begins. This piece of info comes out of The Peoples of Middle Earth.