Immortality

[Tigerlily Gamgee]

Not any Elf can give up their immortality... only Elves that have a mortal relative (mother, father, grandmother, grandfather) can choose to become mortal.
Arwen could give up hers because her father, Elrond, is half-Elven (because of, I think, his mother being mortal...I don't remember exactly, so feel free to correct me if I am wrong)... his brother chose mortality many years prior.
She gave hers up because she wanted to be with Aragorn. She loved him and did not want to live on after he had passed.