Fairy
Fairy is a generic term for any specific member of the Fey (or Fae) races. The iconic fairy is the rootstock from which the modern Sprite and Pixie took form. Other creatures that, in mythology, are indistinguishable from fairies or are part of the fairy family tree include the Elf, the Gnome, the Dwarf, the Brownie, the Leprechaun, the Redcap, the Goblin and the Kobold.
As you might surmise from this, it makes defining what a fairy is to be rather tricky. The generally accepted version is pretty much synonymous with the sprite or pixie; a small, elf-like humanoid with wings - usually butterfly-like, for the more "cutesy" depiction of fairies, although translucent dragonfly-like wings are also quite popular. Their personalities range from sweet and innocent, to fun-loving and carefree, to playful tricksters, to amoral pranksters, hedonists and party-girls.
Fairies in D&D
Dungeons & Dragons has a long, long history with fairies - or faeries, as it prefers to call them. Aside from the iconic Sprite & Pixie, which most other "fairies" are usually described as being subraces of, fairy-like beings have appeared under various names and throughout the editions.
Gorse Faeries are the tiniest known fairy species in D&D - roughly 1/4 the size of the already pretty damn small Atomie. Resembling diminutive elf children with two pairs of dragonfly-like wings, gorse fairies hide in dense, thorny bushes - their namesake is preferred - and just try not to get eaten by the abundance of much bigger nasties that plague the typical D&D world. They have been known to barter with those willing to be non-threatening and to make gifts of milk, honey, bread and fruit, and those who do favors for them can be assured that the gorse fairies won't forget it.
Bramble Faeries are believed to be more malign relatives of the gorse faeries. Flightless, instead having long spines sticking out of their backs, they adorn themselves with carefully worked spiky armor to ward off threats, and are considered the most aggressive and vicious of all faeries. They often tame "spiky" creatures like hedgehogs, porcupines or al'miraj to serve as their steeds. Roughly 1 in 10 of these faeries has the ability to secrete a burning muscle-attacking venom (-2 penalty to all attack & damage rolls for 2d10 rounds, multiple doses have cumulative effects) from their back-spines; these invariably become the leader of their tribes.
Squeakers are distorted, chibi-like spritely faeries that are believed to have been created by a wizard spell that went wrong. They are generally considered a nuisance if they are considered anything at all.
Stwingers are nauseatingly cute faeries that just want to have fun in their own silly, child-like way - mostly by swinging from hair or beards. They actually secrete a pheromone that enraptures anyone who fails to resist it, functioning like a powerful charm spell. This pheromone can be milked from its glands and used as an ingredient to make love potions ("Philters of Love", in D&D terminology).
Faerie Fiddler's are short, almost halfling-like figures known for their masterful skill at playing the fiddle, which gives them almost bardic magical abilities.
The list of D&D faeries is really quite extensive, and goes well beyond the "little elf with wings" archetype. There's a reason that Fey became a creature type from 3rd edition onwards.
Monstergirls
Just like their elfin cousins, female fairies have long been a subject of attention by those whose interests allow for a meeting of fantasy, beauty and xenophilia. The "typical" fairy is a small, winged elf or human, with proportions ranging from human to loli to shortstack. Unusual coloration and antennae stemming from the forehead - which may or may not resemble any specific insect's - are common features, but hardly universal. When portrayed as xenophiliacs, fairies range from coy seductresses to outright hedonists to shy but loving would-be girlfriends. And this is older than you think - Peter Pan's fairy companion Tinkerbell is, in the original novels, basically portrayed as a very possessive and jealous girlfriend.
As for the... logistics... of relationships, that varies widely. Some use magic to either assume human-like stature or to shrink their partner down to their size for trysts - sometimes both. Others are just presented as magically stretchy enough that they can take it despite the size difference. Some are either size-shifters, capable of assuming a form that is still small but not to the point where a human dick is as long as they are tall, or just naturally around the same size as a gnome or halfling, although that's rare. And others just accept that penetration is impossible, instead engaging in very unconventional sexual activities.
In the Monster Girl Encyclopedia, there are currently three known types of fairy mamono; the Fairy, the Leanan Sidhe, and the Titania. There is also a Pixie mamono, but it's actually a kind of Imp that has evolved to look sort of like a Fairy to better lure in human victims.
The standard Fairy is presented as a colorfully butterfly-winged loli elf of diminutive stature. Child-like and carefree, their demonic corruption has turned them from an asexual species into an innocently lustful species. These fit into the "stretchy fairy" mold presented above; they avidly grind themselves against a chosen man until he is aroused, and then force themselves down onto his genitals, feeling intense pleasure as they stretch to fit, effortlessly containing him despite the size difference. They don't have much individual stamina, and can be mastered by someone who keeps his wits, but they often attack in groups.
The Leanan Sidhe is an artistic variant of the common fairy, which draws them to seek out human artists and inspire them to greater proficiency in their chosen art, be it painting, song, poetry, literature, whatever. Since being corrupted into monstergirls, they mostly do this through sex, and it's inevitable that eventually the artist they patronize will become their husband, if not an incubus, and slowly become more concerned with this fae mamono's comfort and amusement than making art. Which is, admittedly, better than the whole "get good but die young" deal that leanan sidhe gave in the real world's mythology, or their vampiric depictions in most other fantasy games.
The Titania is the Queen Fairy species, taking the form of a voluptuous elf woman with large, beautiful colored, butterfly-like wings. Whilst they like to give off the air of being more mature and wise than their "subjects", and see themselves as the caring big sisters of their smaller kin, they're actually quite childlike and very spoiled in nature. They shower their chosen mates with similar compassion and "sisterly love" as their subjects, but also expect a lot of pampering.