The elf-maiden Arwen Undomiel Elelome, youngest daughter of Elrond Half-Elven and Celebrian and grand-daughter to Galadriel and Celeborn, was born in the year 241 of the Third Age. For nearly 3000 years, Arwen lived in Lorien and Rivendell without a worry. The only unhappiness in her earlier years came when her mother was nearly killed by orcs and left for Valinor, leaving her husband and children. Galadriel would later give Arwen her elf-name of Elelome, which means Evenstar, a tribute to her beauty.
During the year 3018, she came across Aragorn and the hobbits Frodo Baggins, Samwise Gamgee, Meriadoc Brandybuck, and Peregrin Took fleeing the Ringwraiths of Sauron. She volunteered to bring an injured Frodo to Rivendell. Since she was the fastest rider, Aragorn let her take Frodo. She rode for many days straight, narrowly avoiding death at the hands of the Ringwraiths, to bring Frodo to her father's land, where he was eventually healed.*
It is there that Arwen confronts Aragorn about his true heritage as ruler of Gondor.* Aragorn isn't willing to listen and Arwen hands him her elf-star, saying that she chooses him as her true love, thus making herself mortal. This takes Aragorn by surprise, because Elrond had said he could not have his daughter's hand until he could claim to be the King of the West again. Doing so, however, put a chasm in Arwen's heart, because by doing so, she has chosen the humans over the elves.
Much later, after the Fellowship of the Ring collapses, Arwen gives a banner to Aragorn when he leads the Forces of the West. She had spent countless hours working on that banner while he was gone, going through the motions of smiling and laughing for her people. She eagerly awaited news of the battle and eventually learned that Aragorn and his companions were victorious and that the One Ring had been destroyed in the fires of Mount Doom.
Aragorn and Arwen wed in Minas Tirith on Midsummer's Day. Sadly, she said goodbye to her father, who was leaving Middle Earth. Aragorn and Arwen would go on to have one son, Eldarion, and five daughters. She was so happy that she forgot about Aragorn's mortality - the shock of his death was almost too much for her. In the year 121 of the Fourth Age, she journeyed to the fading beauty of Lorien and died under its trees the following winter.
Played by the lovely Liv Tyler in the movie series.
From The Lord of the Rings by J.R.R. Tolkien
* These events are according to the movie version (thanks yam); in the books, it was Glorifindel who drowned the Ringwraiths and Arwen and Aragorn never had this talk about his heritage.
In general, it appears that there was more romance added to the movie than there was in the books. I am sorry about my earlier incorrect implications - I myself have not read the series for 8 years so I guess I was swayed by the movie. I'd like to thank yam for pointing out what was wrong in my writeup. I believe everything up there is correct and stuff taken from the movie is cited as such.