Half-elven

The Half-elven (Elvish singular Peredhel, plural Peredhil), are the children of the union of Elves and Men.

The First Age
Two important marriages in the First Age of Middle Earth resulted in the mixing of elvish and mortal blood.

The first of these was between the mortal Beren of the House of Bëor and Luthien, daughter of the elf Thingol, king of the Sindar and the Maia Melian. Beren died in the quest for the Silmaril, and in despair, Luthien'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 deaths.

Their son Dior was thus one-quarter elvish by blood, although since he was born after their rebodiment, the nature of his mortality is quite unclear. He died young in any case, when the sons of Fëanor attacked Doriath.

Dior's wife was Nimloth, a Sindarin elf, and with her he had three children: Elwing, Eluréd, and Elurín. Eluréd and Elurin were slain along with Dior, 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 and Idril of the Noldor and Vanyar. Their son was Eärendil. After the fall of Gondolin, Eärendil also escaped to the Mouths of Sirion, and married Elwing. They had two sons, Elrond and Elros.

The Choice of the Half Elven
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, and granted eternal life in the Undying Lands; or to be counted as mortals, and granted the ineffable Gift of Men. This choice could be delayed, although not indefinitely.

Eärendil would rather have chosen the kindred of Men, but he chose the Elves for his wife Elwing's sake, who chose the Elves. Eärendil would thereafter forever sail 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 return.

Elros chose to be counted among mortals, and became Tar-Minyatur, the first king of Númenor. He finally took his death at the age of five hundred years. The heirs of Elros were not given this choice, but their lifespan was enhanced many times that of normal Men, and they had the freedom to take it when they willed. 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, serving the household of Gil-galad until the end of the second age, and founding Rivendell in the third. He married the elf Celebrian, daughter of Celeborn and Galadriel, and sailed into the west at the conclusion of the War of the Ring.

The children of Elrond and Celebrian were also given free choice of kindred, and therefore Arwen could choose to be counted amongst the Edain even though her father had chosen to be counted as Elven. She exercised this option, marrying Aragorn II Elessar, king of the Reunited Kingdom, and finally dying at the age of 2,901 years. Their son Eldarion and their daughters were not counted as Half-elven, but rather as Dúnedain restored.

We are not sure of whether Elladan and Elrohir chose to be of the Edain or the Eldar; this information is not given in the books. However, since their choice was expressed by whether they would follow their father to Valinor at the time of his own departure, and they were described as remaining in Rivendell, they are often said to have chosen the Edain.

Other Lines
It was a tradition in Dol Amroth that Imrazôr the Númenórean had married an Elf, and therefore the Princes of Dol Amroth were of Elven descent. Legolas of Mirkwood believed as much upon meeting Prince Imrahil, but the matter remains unclear.

Beyond Tolkien
Unlike in some other fantasy worlds, Tolkien's Half-elven are not a distinct race per se. In Dungeons & Dragons and other fantasy role-playing games, these beings are simply known as half-elves (singular half-elf) instead of using other terms. Some other fantasy stories call beings who are half human and half another race (usually an elf) halflings; the term in Tolkien's legendarium pertains only to hobbits.