In the histories of Middle-earth, there were a very few figures who shared descent from both Elves and Men. There were three definite cases where a Man and an Elf produced half-elven children: those of Beren and Lúthien, Tuor and Idril; and Aragorn and Arwen. There's also a possible fourth case, a little more obscure than the others: that of Imrazôr and Mithrellas.
As a title, Half-elven (Elvish Peredhil) is normally only applied to Elros and Elrond, the grandchildren of Tuor and Idril. Like their parents Eärendil and Elwing, these were each allowed to choose their own destinies. Elros chose to live as a Mortal Man, and became Tar-Minyatur, the first King of Númenor. His brother Elrond chose the other path, to be considered an Elf, and remained in Middle-earth until the end of the Third Age.