In The Lord of the Rings, Arwen (an Elf maiden) and Aragorn (a human man who was one of the Dúnedain—a descendant of the Númenóreans), were in love. Though the Dúnedain were blessed with long life and slow aging, they were still mortal. On the other hand, Elves are immortal beings that never succumb to old age. They can even recover from wounds that would be fatal to Man. Although they can be killed in battle, all Elves' spirits go to The Halls of Mandos when they die.
In the books, Arwen possesses a stone called the Elfstone, which was a green gem on an eagle-shaped brooch that had the light of the Sun within it. Looking through the gem would cause the viewer to see all that was now old as young and new again. But in the movies, this stone was replaced by the Evenstar (which was not in the books, except as a nickname for Arwen) to show how Arwen was one of the last Elves to live in Middle Earth, and she was as bright and light as the star. The Evanstar was most likely used to replace both the Elfstone and the "white gem like a star" on a silver chain that Arwen gives to Frodo after he finally destroys the Ring as a gift to comfort him through troubling times (in the books).
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But in the movies, the Evenstar was given to Aragorn by Arwen to show how she would forsake her immortal life to be with her one true love—saying, "I would rather spend one lifetime with you than face all the ages of this world alone." Many first-time viewers might have thought that this necklace was the source of Arwen's immortality, and that by giving it to Aragorn, she was giving him an immortal life and giving up her own. Aragorn even says "you cannot give me this" to which Arwen replies "it is mine to give to whom I will, like my heart." But this is not the case, and the necklace is only symbolic of Arwen's decision to stay in Middle Earth with Aragorn and not sail to the Undying Lands with her people.
So even though giving away this necklace was not Arwen giving away her immortality, it was symbolic of it. Did Arwen lose her immortality? The answer is yes. Arwen's father was Elrond. Elrond has only Half-Elf, and he was once given the choice to be counted among Man or Elves. When he chose to be counted as Elven kind, he was then given immortality. Because Arwen is Elrond's daughter and also Half-Elf, she too has the choice to be a mortal woman or an immortal Elf. In Arwen's mind, there is nothing that can separate her from Aragorn, as she would rather die happy with him than regret her decision to leave him for all of eternity. This is especially so when she has a vision of the child they would have together if she stayed.
When Arwen chose to stay in Middle Earth because of the hope she had for a brighter future, she also chose to be counted among the race of Man. Elrond confronts Aragorn in The Return of the King and says that Arwen is dying, her fate tied to the Ring. Arwen isn't necessarily dying because of the Ring, but now that she is mortal, she is dying through the slow decaying of time. Arwen also faces the same fate as all those in Middle Earth should the Ring Bearer fail his mission. So in that way, her fate is tied to the Ring. Elrond believed the world of Men would soon fall and that there was no hope, so he chose not to fight in their battle.
It is only after Arwen makes her decision to become mortal that Elrond mended the blade that cut the Ring in The Reforging of Narsil. This sword would allow Aragorn to command those that dwell in the mountains—the Army of the Dead—so that the Free Peoples of Middle Earth would have a fighting chance. If Arwen had not stayed, Elrond would not have reforged the blade, nor would he have confronted Aragorn and told him to gain command of the dead. Elrond helped not for Man or even all of Middle Earth, but for Arwen since she would be doomed too if they should fail and Sauron should regain control.
Though Arwen doesn't get much screen time, she is perhaps the reason the war took a turn in favor of Mankind. Without her decision to stay with Aragorn, her one true love, The Battle of the Pelennor Fields might have been lost. This battle was the last battle before The Battle of the Morannon (fought in front of the Black Gates in The Lord of the Rings: Return of the King) and without its victory, Frodo and Sam would not have had the needed distraction from The Great Eye in order to complete the mission.
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