Sphinx
The Sphinx (plural: sphinges, but it's rarely used) is a mythical monster from Egypt, Greece and the Asiatic regions of Earth, taking the form of a large cat (lion, specifically) with the head (and tits, if female) of a human, and sometimes with eagle wings. It has also appeared in a number of fantasy settings. Because the most famous form of the sphinx is the Greek one, almost always depicted with big tits to go with her human face, she is often held up as one of the real-world monstergirls, though, like centaurs, female sphinxes do elicit complaints of "too furry!" since, y'know, you're fucking what is basically a she-cat with a human's head.
Mythology
The Egyptian sphinx, from little we know, is always portrayed as a lion (to symbolize strength) with the head of a human (symbolizing intelligence); with these combined aspects, the sphinx as a whole was often used to symbolize power. Rather like the Lamassu and Shedu of Sumeria, it is a protector spirit. There are several kinds of sphinx from Egypt; the human-headed ones, falcon-headed ones (hieracosphinxes) and ram-headed ones (criosphinxes).
The Greek sphinx is the most famous version. This is the version that sometimes incorporates eagle wings, and is always female. Set itself up on a rock leading to Thebes and demanded that people guess the answer to her riddle or be eaten. If they got it wrong, she ate them anyway. Finally, Oedipus answered it correctly, whereupon she got pissed and killed herself. Because of how popular Greek myths are, everybody's heard the riddle - "what goes on four legs in the morning, two legs at noon and three legs in the evening?" For those of you who've been living under a stump, the answer is "man", since he crawls on all fours as a baby (morning), walks upright as an adult (noon) and uses a walking stick as a third leg as an old geezer (evening).
The Asiatic sphinx, like the Egyptian sphinx, is a protector spirit, but may have wings like the Greek sphinx.
D&D and Pathfinder
Okay, to begin with, one has to remember that spinxes appeared in Advanced Dungeons & Dragons, when monster manuals liked to try and be "realistic", but tended to fail hilariously - witness the absurdity of medusa/maedar mating outcomes. D&D basically throws together the Greek and Egyptian sphinxes, ending up with four seperate kinds:
- Gynosphinxes are your standard Greek sphinx with some Egyptian thrown in. Winged lionesses with human heads and tits. Highly intelligent, love riddles, natural spellcasters. Drawn to guarding sacred spots. Don't usually eat people, but won't hesitate to do so if hungry, bored or insulted, especially if a person is a moron.
- Androsphinxes are the male, more good-natured equivalents to gynosphinxes. Even smarter and more magically powerful. Don't like company much, and pretty prudish.
- Criosphinxes are fairly dimwitted ram-headed winged lions. Some minor druidic abilities, but mostly just beasts.
- Hieracosphinxes are savage, rapacious, totally evil eagle-headed winged lions.
So, here's the weirdness: only gynosphinxes are female, and all others are male. Gynosphinxes are only born to androsphinx fathers - but androsphinxes don't like sex. So you have a species where the females desperately try to get laid from the one set of menfolk that can give them daughters, whilst the other sets have to rape the womens if they want to have sons. Seriously, the fuck? Was this cooked up by bitter nerds with a furry streak or some shit?
5e messed up the communication between the artist and the designers, getting a Mufasa-looking Androsphinx and a Gynosphinx with a clearly lion-looking head despite the text describing them as having humanoid heads. In this edition sphinxes are fucking hardcore when encountered in their lairs (though fortunately these abilities can be used only once each before requiring a short rest: they can force you to reroll your initiative as long as you're there: take a DC 15 save or age or de-age 1D20 YEARS, creating an interesting case of "Adventurer babies wat do?" (healed by Greater Restoration, which the Androsphinx has), send you up to a decade back or forward in time (Wish to return) or shift itself and up to seven creatures it can see to another plane, then return as a bonus action. So in other words, if you piss a Sphinx off it can just drop your ass into the fucking Abyss. Have fun swimming in The Gaping Maw, chucklefuck! Say hi to Demogorgon for me!
Pathfinder basically keeps the same setup, but makes the species of sphinx depend on the relationship between parents (love causes andro/gyno mixed gender fraternal twins, lust causes crio, rape causes hieraco) rather than father's species and adds a fifth race, the jackal-headed malevolent necromancer crinosphinxes.
Warhammer Fantasy
Sphinxes are not known to have actually existed as a living creature in Warhammer, and instead the Tomb Kings (an Egyptian-expy race of living skeletons) of Nehekhara (not-Egypt) created living statues of them. They come in two varieties, the Necrosphinx and the Khemrian Warsphinx. The former possesses wings and giant blade weapons resembling a second pair of wings, and bears the face of a man of some importance in history (the older ones have human faces, those created after the death and debirth (unbirth? antibirth?) of the Nehekharans have skeletal ones, and its very possible for the same Tomb King to thus have a Necrosphinx depicting his face before and after death). The latter resembles a skeletal Sphinx and has a Howdah (a carriage mounted to the back of a large animal) in which either an important person within the Tomb Kingdom rides or else a team of highly trained warriors bearing spears.
MGE
In the Monster Girl Encyclopedia, sphinxes are dusky-skinned, blond-furred, brown-haired catgirls (goddamit, KC! Way to make the case for your critics who call your girls cosplayers!) who live in the desert region. They ask riddles of men; those who win get pounced upon and fucked, those who fail are charmed by her magic to try and compel them to fuck her anyway.