Fey
Fey, also spelled as Fae, is a generic term used to refer to any creature associated with the fairy races, such as sprites, pixies, nymphs, etcetera. The term is pretty much as old as the concept of the Land of Faerie itself, but in /tg/ circles was popularized by Dungeons & Dragons, which designated it as one of the basic creature types in 3rd edition.
In Mythology
The list of fey monsters is vast indeed, and includes:
In D&D
Virtually every fey creature that appeared in mythology has appeared in Dungeons & Dragons at some point or another. Fey occupy a weird nebulous point between being magical humanoids and being full-fledged extraplanar beings, like the angels, fiends and elementals. In the World Axis cosmology, and 5e's modified Great Wheel, fey became full-fledged extraplanar beings with their own home plane, the Feywild.
The rulers of the fey are called, imaginatively enough, the Archfey.
Fey creatures often have names preceded by "fairy/faerie", such as the Faerie Dragon.
Shadow Fey
Introduced in the D&D setting of Ravenloft, Shadow Fey are a class of fey creatures who escaped servitude to a monstrous demon-god called Gwydion in the Plane of Shadow by fleeing into the Demiplane of Dread. They are basically a more "Gothic themed" variation of the classic fey creatures of D&D at the time, and are characterized predominantly by their lethal aversion to sunlight and their active embrace of the classic Seelie/Unseelie mythology split, as they stem from a time when fey creatures tended to just sort of "there" in D&D.
For more details, see the Shadow Fey page.