Wikipedia
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 (they do not die, or if killed are re-embodied, while Arda endures), whereas Men are mortal (after death their souls depart the world for 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.