In Lord of the Rings, three thousand years before the story, Elrond is seen fighting in the war against Sauron. But after three thousand years he is still seen young. So I am guessing that elves are immortal.
But in the battle of Helm's Deep many elves die fighting Saruman's army. So they can't be immortal.
So are they mortal or immortal? How can Elrond's not aging be explained?
They can't die by age, but they can die by the sword or grief.
According to Tolkien, once an Elf becomes an adult, they stop getting older. They are also less vulnerable against physical damage, but they aren't immortal. The lives of Elves only endure as the world endures.
Elves could be slain or die of grief (their spirit leaves their body), but were not subject to age or disease. When an Elf dies, his spirit goes to Mandos for his judgement, and after a period of waiting could be reembodied.
According to Wikipedia
Elves are naturally immortal, and remain unwearied with age. In addition to their immortality, Elves can recover from wounds which would normally kill a mortal Man. However, Elves can be slain, or die of grief and weariness.
Spirits of dead Elves go to the Halls of Mandos in Valinor. After a certain period of time and rest that serves as "cleansing", their spirits are clothed in bodies identical to their old ones. However, they almost never go back to Middle-earth and remain in Valinor instead. An exception was Glorfindel in The Lord of the Rings; as shown in later books, Tolkien decided he was a "reborn" hero from The Silmarillion rather than an individual with the same name. A rare and more unusual example of an Elf coming back from the Halls of Mandos is found in the tale of Beren and Lúthien, as Lúthien was the other Elf to be sent back to Middle-earth – as a mortal, however. Tolkien's Elvish words for "spirit" and "body" were fëa (plural fëar) and hröa (plural hröar) respectively.
Interesting info from here
While the three cycles are not specifically defined, the first cycle is likely childhood and adolescence, which ended at the 100th year, the second is adulthood which could continue for Ages, and the third is for extremely old Elves; Elves did not physically age after they reached maturity, but they did age in a different sense than Men. They became ever more weary of the world and burdened by its sorrows. Elves are naturally immortal; like the Ainur, they are bound to Arda until its End. Elves are immune to all diseases, and they can recover from wounds which would normally kill a mortal Man.
The only "unkillable beings" in LOTR are the soldiers of The Army of the Dead. The Army of the Dead was cursed by Isildur with immortality in the form of an existence as undead skeletons after they abandoned their oath in the War of the Last Alliance. Their only chance of dying was to fulfill their oath.
Yes they are immortal until killed:
As told in The History of Middle-earth and in Tolkien's Letters, Elves had a different life cycle from Men. Most of the following information strictly refers only to the Eldar, as found in his essay Laws and Customs among the Eldar, found in Morgoth's Ring.
Elves are born about one year from their conception. The day of their conception is celebrated, not the actual birthday itself. Their minds develop more quickly than their bodies; by their first year, they can speak, walk and even dance, and their quicker onset of mental maturity makes young Elves seem, to Men, older than they really are. Physical puberty comes in around their fiftieth to one hundredth year (by age fifty they reach their adult height), and by their first hundred years of life outside the womb all Elves are fully grown. Elven bodies eventually stop aging physically, while human bodies do not
Elves are naturally immortal, and remain unwearied with age. In addition to their immortality, Elves can recover from wounds which would normally kill a mortal Man. However, Elves can be slain, or die of grief and weariness. - wikipedia
Elrond helpfully explains the consequences for this if Arwen binds herself to Aragorn, a human. He points out that Aragorn will age and die, but Arwen will linger on.
ELROND: If Aragorn survives this war, you will still be parted. If Sauron is defeated and Aragorn made king......and all that you hope for comes true......you will still have to taste the bitterness of mortality.
A vision of Arwen dressed in mourning robes appears. She is standing beside Aragorn, who lies grey and dead wearing his crown and grasping his sword on top of his tomb. Mourners walk behind her.
ELROND : Whether by the sword or the slow decay of time......Aragorn will die. And there will be no comfort for you......no comfort to ease the pain of his passing. He will come to death......an image of the splendor of the kings of Men......in glory undimmed before the breaking of the world.
ELROND: But you, my daughter......you will linger on in darkness and in doubt......as nightfall in winter that comes without a star. Here you will dwell......bound to your grief under the fading trees...... until all the world is changed.....and the long years of your life are utterly spent.
The Elves are leaving Middle Earth on their ships for Valinor. Any Elves that remain indefinitely will eventually be overtaken by a changing world in which their existence no longer makes sense.
ELROND: Your hands are cold. The life of the Eldar is leaving you.
ARWEN: This was my choice. Ada, whether by your will or not there is no ship now that can bear me hence.
Transcripts at AgeOfTheRing.com: The Two Towers and The Return of the King.
