In J. R. R. Tolkien's fictional universe of Middle-earth, the Half-elven (Sindarin singular Peredhel, plural Peredhil, Quenya singular Perelda) are the children of the union of Elves and Men. The Half-elven are not a distinct race from Elves and Men, and must ultimately choose to which race they belong (or at least, so must the half-elf descendants of the couple of half-elves that reached Valinor). This is significant because although Elves and Men are able to crossmate and produce fertile offspring, their final fates are separate: Elves are immortal (while the world endures, they do not die of natural causes, and may be reembodied if killed), whereas Men are mortal (after death their souls depart the world to an unknown place and future).
There were three, possibly four, couplings of Elves and Men that generated descendants. Three of these unions occurred between members of the foremost dynasties of Elves and Men, the High Elves and the Edain: They were Idril and Tuor, Lúthien and Beren, Arwen and Aragorn. The first two couples wed during the final part of the First Age of Middle-earth while the third married at the end of the Third Age (some six thousand-five hundred years later). The third couple descended not only from the first two couples, but also from the twin Peredhil, Elros and Elrond, who chose mankind and elvenkind as their respective races—thereby severing their fates and those of their descendants. In Appendix A of The Return of the King, Tolkien notes that by the marriage of Arwen and Aragorn "the long-sundered branches of the Half-elven were reunited and their line was restored." The second union was the only one of the three marriages in which the Elf involved (Idril) did not become mortal; instead Tuor was joined to the Elves. In all four known cases, the husband was mortal, while the wife was Elven.
The fourth coupling, between Imrazôr and Mithrellas, was the traditional founding of Prince Imrahil's line.
Like many other ideas in Tolkien's mythos, the notion of half-elves is borrowed from Norse mythology, in which elves occasionally had children with humans.
Two important marriages in the First Age of Middle-earth resulted in the blending of Elvish and mortal blood.
The first of these was between the mortal Beren, of the House of Bëor, and Lúthien, daughter of the Elf Thingol, king of the Sindar, and Melian, a Maia. Beren died in the quest for the Silmaril, and in despair, Lúthien's spirit departed her body and made its way to the halls of Mandos. Mandos allowed them a unique fate, and they were re-bodied as mortals in Middle-earth, where they dwelt until their second deaths.
Their son Dior, heir of the Sindarin kingdom of Doriath and of the Silmaril, was thus one-quarter Elvish by blood and one-quarter Maian (thus half-immortal), and half-human (thus half-mortal). Since he was born after his parents' re-embodiment, however, the nature of his mortality is unclear. In any case, he was killed while still young, when the sons of Fëanor sacked Doriath.
Dior's wife was Nimloth, a Sindarin Elf, and with her he had three children: Elwing, Eluréd and Elurín. Eluréd and Elurín were slain along with Dior — or escaped, never to be heard of again, while Elwing escaped to the Mouths of Sirion.
The second marriage of Men and Elves in the First Age was between Tuor of the House of Hador, another branch of the Edain, and Idril Celebrindal, an Elf, though half Noldorin and half Vanyarin in ancestry. Their son was Eärendil. After the fall of Gondolin, Eärendil also escaped to the Mouths of Sirion, and married Elwing. They had twin sons, Elrond and Elros. Both sons are one sixteenth Maiar, nine sixteenths elven (five thirty-seconds Vanyarin, three thirty-seconds Noldorin, five sixteenths Sindarin) and three eighths human (one quarter of the House of Bëor, one sixteenth of the House of Haleth, and one sixteenth of the House of Hador).
After the conclusion of the War of Wrath, Manwë determined that the surviving Half-elven would have their choice of fates: to be counted as Elves (free to dwell in the blessed Undying Lands for as long as Arda endures) or to be counted as Human (entitled to the Gift of Men whereby, through death, their spirits are freed to enter the unknown beyond Arda). This choice could be delayed, although not indefinitely.
Eärendil would rather have chosen the kindred of Men, but Elwing preferred elvenkind. Moreover, having sailed to the Undying Lands with the power of the Silmaril, Eärendil was not permitted to return to mortal lands. Thereafter he was set aloft, to sail forever the heavens in his ship Vingilot, the Silmaril of Beren and Lúthien on the prow. In Middle-earth, he was seen as the evening star, and the light of his Silmaril was captured in the Phial of Galadriel. Elwing built a tower in the Shadowy Seas and often met him on his daily return.
Elros chose to be counted among mortals, and became Tar-Minyatur, the first king of Númenor. He finally took his death (for those kings had the freedom and grace to die at will) at the age of five hundred. The descendants of Elros were not given this choice, but their lifespan was enhanced several times that of ordinary Men. In later times the Númenórean kings, descendants of Elros, regretted their forefather's choice, and this helped lead to the Downfall of Númenor.
Elrond chose to be counted among the Elves, joining the court of Gil-galad until the end of the Second Age. He also founded Rivendell — haven of the Peredhil — in the Second Age. He married the Elf Celebrían, daughter of Celeborn and Galadriel, and sailed into the West at the conclusion of the War of the Ring.
The children of Elrond were also given choice of kindred, and therefore Arwen could choose to be counted among the Edain even though her father hoped she would accompany him to Elvenhome in the West. But she chose otherwise, marrying Aragorn II Elessar, king of the Reunited Kingdom, and died alone at the age of 2,901 years, grieving the brevity of her mortal happiness. Their son Eldarion and their daughters were not counted as Half-elven, but rather as Dúnedain restored.
It is not stated in Tolkien's books whether Arwen's brothers, Elladan and Elrohir, chose to be of the Edain or the Eldar. But their decision, too, was to be manifested by accompanying their father over the sea at the time of his own departure — or not. Yet they are described as remaining at Rivendell, so some readers conclude that they exercised their right to live and die in Middle-earth as Edain.
It was a tradition in Dol Amroth that Imrazôr the Númenórean had married an Elf and therefore his descendants, the Princes of Dol Amroth, were of Elven descent. Legolas, an Elf of Mirkwood, believed as much upon meeting Prince Imrahil, but the matter is probed no further in The Lord of the Rings. In Tolkien's Unfinished Tales, however, Imrazôr's wife in one account is given as Mithrellas, handmaiden of Nimrodel, a Silvan Elf who resisted the encroachment of the Eldar in her homeland, Lothlórien.
In The Hobbit reference is made to a rumour among Hobbit folk that a Took ancestor of Bilbo Baggins had taken a "fairy" (i.e. Elf) wife, but the allegation is immediately dismissed as a simplistic explanation for the sometimes atypical behaviour of the Took clan.
In The Book of Lost Tales (published in two parts), the young Tolkien originally intended Eärendil, then spelled Earendel, to be the first of the Half-elven. Early versions of the tale of Beren and Lúthien had Beren as an Elf. The earliest version of the tale of Túrin had Tamar, the character Tolkien later renamed Brandir, as a Half-elf; Tolkien mentioned this in a way that implied he did not consider Half-elven descent especially remarkable at the time he wrote that story.Tolkien, J. R. R. (1977), Christopher Tolkien, ed., The Silmarillion, Boston: Houghton Mifflin, ISBN 0-395-25730-1
Tolkien, J. R. R. (1984), Christopher Tolkien, ed., The Book of Lost Tales, 1, Boston: Houghton Mifflin, ISBN 0-395-35439-0