There were at least four couplings of Elves and Men which generated descendants. Three of these unions occurred between members of the foremost dynasties of Elves and Men, respectively, the High Elves and the Edain: They were Lúthien and Beren, Idril and Tuor, and Arwen and Aragorn. The first two couples wed during the final part of the First Age of Middle-earth whilst the last married at the end of the Third Age (some six thousand-five hundred or so years later). The last couple descended not only from the first two couples, but also descended from the twin Peredhel, Elros and Elrond, who chose mankind and elfkind as their respective heritages -- 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 have to become mortal, but rather Tuor was joined to the Elves. In all four known cases, the husband was mortal, while the female was Elven.
In the fourth elf/human coupling, alluded to only indirectly in the trilogy, it is not clear that the ancestress (Mithrellas) of Imrahil, Prince of Dol Amroth belonged to one of the Elven kindreds entitled to the immortality of those who dwell in the Undying Lands, although her husband (Imrazôr) was a man of the Edain kindred.
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 inheritor of the Nauglamír gilded 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 fourth of the House of Bëor, one sixteenth of the House of Haleth, and one sixteenth of the House of Hador).
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, serving in the household of Gil-galad until the end of the Second Age, and founding Rivendell—haven of the Peredhil—in the Third. 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 free choice of kindred, and therefore Arwen could choose to be counted amongst 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 were described as remaining at Rivendell, so some readers conclude that they exercised their right to live and die in Middle-earth as Edain.