Half-elves were crossbreeds between and elf or any other being. They were called so because of their extremely mixed Adan/Edain (Man) and Eldarin (Elven) blood. At the end of the First Age, they were given a choice to be Elven and immortal, or to be of the race of Men and accept the Gift of Atani.
First AgeEditTwo important marriages in the First Age of Middle-earth resulted in the mixing of elvish and mortal (Edain) blood.
The first of these was between the mortal Beren Erchamion of the House of Bëor and Luthien, daughter of the elf Elu Thingol, the King of Doriath and High King of the Sindar and his Queen Melian the Maia. 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 embodiment. It is quite unclear that he is a full-blooded Atani since due to the nature of his mortality. 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. During the Sack of Menegroth by the Sons of Fëanor, Elwing escaped to the Mouths of Sirion while Eluréd and Elurin were seized by servants of Celegorm and abandoned in the forest to die. They were never found and were presumed to have perished in the forest.
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.
During the Third Age, Galador, the son of Imrazôr (Atani) and Mithrellas (Elf), led a colony of Half-elves that chose to live a mortal life while those that moved to Rivendell, chose to remain immortal. After Galador became the First Prince of Dol Amroth, the mortal half-elven population began to increase over the centuries. However, some mortal half-elves have chosen to fall in love with the Atani that lived in Gondor and give birth to the full-blooded.