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In the movie, "The Fellowship of the Ring," Arwen gives Aragorn a pendant called "the Evenstar. Contrary to what most movie-goers believe, the Evenstar does not hold Arwen's immortality. It is merely a gift to Aragorn so that he may remember Arwen, and think of her when he looks upon it. In the book, "The Fellowship of the Ring," that event was not mentioned. Though, the Evenstar was given to Aragorn.
Arwen is given the choice to either pursure her dream, and marry the one she loves, or to go and be with her people in the Undying Lands. That choice is very hard, and it frustrates her very much. But either way, she will be filled with regret.
Whether she makes the right choice, remains hidden.
In Peter Jackson's first movie, Arwen saved Frodo from the wound of the witchking. The witchking is the leader of the ringwraiths. Though, in J.R.R. Tolkien's books, this was not so. An elf named Glorfindel saved him. Glorfindel brought Frodo to Rivendell on his horse Asfaloth.