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.
The image actual medieval Europeans held of the Fae (or "The Fair Folk" as they also called them so as to not earn their ire, hence the word fairy) was nothing like the glittery friendly gits that Disney raised you on. The various fae were generally Chaotic Neutral at best and Chaotic Evil at worst, doing stuff like abducting your newborn child at night and replacing them with their own or burning your crops to the ground for the lulz. As otherworldly inscrutable beings who did with mortals as they pleased, benign ones were uncommon and benevolent ones rare. That being said, much like devils or to some extent djinn, the Fair Folk in the stories tend to follow strict rules, placing great value on oaths and agreements, but tend to follow things to the letter and not the spirit. There are many tales were mortals make bargains with one of the Fair Folk, only to get screwed over because they worded their request badly. Usually, the stories make it very clear that under no circumstances should you offend them because the Fey's version of mercy tends to involve a lot of pain and suffering.
Of surprising note is the fact that folks would often report "Fey abduction" encounters that had all the same characteristics as modern "Alien abduction" accounts; just replace the Greys and Reptiods with elves and goblins, and the spooky spaceship with some spooky cave or forest.
Also like abducting aliens, most medieval people didn't actually believe in them. In fact, St Agustine was once asked how they fit into the Genesis Creation story (which would actually be a really good question, come to think of it), and he responded by saying that they can worry about that once their existence had been definitively proven.
The list of fey monsters is vast indeed, and includes:
Virtually every fey creature that appeared in mythology has appeared in Dungeons & Dragons at some point or another, sometimes as Medieval Europeans imagined them, sometimes as Disney imagined them, sometimes somewhere in between. Fey occupy a weird nebulous point between being magical humanoids and being full-fledged extraplanar beings, like the angels, fiends and elementals. The Plane of Faerie as the otherworldly home realm of the fey has been thrown around a few times in the Great Wheel, but it wasn't until 4e introduced the World Axis cosmology that this became an integral part of the basic multiverse, known as the Feywild. This went down well enough that WotC thought it was worth risking grognard backlash by incoporating the Feywild into a modified Great Wheel in 5e.
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.
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.
Even excluding the obvious Elves, many of the various Fey races have been made playable in D&D at one point or another, particularly in the days of Basic and Advanced D&D. However, in one of the more confusing uses of the name, issue #257 of Dragon Magazine presented the Fey as a kind of alternate Elf race for use in its "Dark Ages" mini-setting; a Low Historical Fantasy version of Dark Ages Britain running off of the Advanced Dungeons & Dragons ruleset.
"Fey elves", visually characterized by their blonde to red-blonde hair and piercing green eyes, function almost exactly like standard AD&D elves, except for a few minor changes:
