Although there may have been many such, only three are named as "Half-elven":
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's fate was special however: he was not allowed to stay in Valinor, but had to sail the heavens in his ship Vingilotë, the Silmaril of Beren and Lúthien on the prow, as a sign of hope for Middle-earth. Elwing built a tower in the Shadowy Seas and often met him on his return.
The heirs of Elrond, including Arwen Undómiel, also had the free choice of kindred, therefore Arwen could choose to be counted amongst the Edain even though her father had chosen to be counted as Elven. The heirs of Elros were not given this choice, but their lifespan was enhanced many times that of normal Men. In later times the Númenórean Kings, descendants of Elros, regretted their forefather's choice, and this helped lead to the Downfall of Númenor.
Dior was the son of Beren and Lúthien, and he was therefore also counted as Half-elven, as were his children with the Elf Nimloth of Doriath, Eluréd, Elurín, and Elwing. Eluréd and Elurín were killed, but even though Elwing was given her choice of kindred she nor her brothers or father are called Half-elven by Tolkien.
Arwen, daughter of Elrond married Aragorn heir of Elros, and thus the two Half-elven lines were merged in the Fourth Age. Their son Eldarion and their daughters were not counted as Half-elven, but rather as Dúnedain restored.
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, but they were not counted as Half-elven.