Arwen Undomiel (Undomiel means "Evenstar" in Sindarin) was born in 241 TA. She was a daughter of Elrond, the Elf-lord, and Celebrian. She was so charming and beautiful that she resembled Luthien Tinuviel, who was "the fairest maiden that has ever been among all the children of this world".
According to the book we meet Arwen at the Council of Elrond, however in the film she appears a little bit earlier. Anyway, this is how this wonderful elven maiden appeared before Frodo: "...there was a chair under a canopy, and there sat a lady fair to look upon, and so like was she in form of womanhood to Elrond that Frodo guessed that she was one of his close kindred. Young she was and yet not so. The braids of her dark hair were touched by no frost, her white arms and clear face were flawless and smooth, and the light of stars was in her bright eyes, grey as a cloudless night; yet queenly she looked, and thought and knowledge were in her glance, as of one who has known many things that the years bring. Above her brow her head was covered with a cap of silver lace netted with small gems, glittering white; but her soft grey raiment had no ornament save a girdle of leaves wrought in silver." At first Frodo couldn't believe his eyes that such beauty existed in Middle-earth. Little hobbit felt a bit frightened, for he was honoured to sit next to Elrond and his daughter at this special conference calling together what would become the members called the Fellowship of the Ring.
Almost three thousand years the elven maiden spent in her father's home, occasionally she visited her "always young" grandmother, Galadriel, until she met Aragorn in 2951. The king-to-be of Gondor fell in love with Arwen. For Aragorn she became that fair lady, for the sake of whom every true knight performed his deeds. In her thoughts Arwen was constantly with her beloved. No, she didn't go to the fields of battle with a sword, but sent Aragorn a splendid standard of Gondor, which she wrought in secret. And asked to pass him the following words: "The days now are short. Either our hope cometh, or all hopes end. Therefore I send thee what I have made for thee. Fare well, Elfstone!"
After the fall of Sauron, "Aragorn, the King Elessar, wedded Arwen Undomiel in the City of the Kings upon the day of Midsummer, and the tale of their long waiting and labours was come to fulfilment."
It was incredibly hard to part with Elrond, her father, for she was now a mortal woman by her own will. She told Frodo that she admired Luthien who, like she, had chosen to become mortal for the sake of love. And her choice was both "the sweet and the bitter". She offered the Ring-Bearer to sail into the West, where he would be able to heal his wounds and weariness. Also she gave Frodo a white gem that looked like a star, and said: "When the memory of the fear and the darkness troubles you, this will bring you aid."
Arwen gave her husband several daughters and a son — the heir to the throne. She died in 121 FA a year after her husband passed away.
Arwen was the epitome of beauty and the symbol of a fair lady.