Catfolk
/tg/ and the furry fandom have long had a great deal of hostility towards each other. And yet, hate and love can be said to be two sides of the same coin. Furries have often shamelessly involved themselves in roleplaying games, from the likes of Ironclaw and its weeaboo spin-off Jadeclaw to Teenage Mutant Ninja Turtles And Other Strangeness. And yet, even more mainstream games have thrown them some encouragement.
Cats are, without a doubt, one of the most commonly admired of animals by furries. Consequently, anthropomorphized cats are a common feature in fantasy and science fiction settings, even in games; ironically, catgirls, despite being monstergirls, are very much a rarity in official game materials, because they are seen by many writers as being "too deviant and/or weeaboo". Yet catfolk somehow get a pass on this...
Anyway, there's a lot of different cat-people races that run around various game settings. This article will serve to house and refer to them all.
Dungeons & Dragons
Catfolk in Dungeons & Dragons go all the way back to first edition, first appearing in the modules X2: Castle Amber and X1: Isle of Dread, alongside the Lupin (dog/wolf-people, another hugely common furry race).
These first edition catfolk were called the Rakasta. They got promoted to playable character in Champions of Mystara, and were the main identity for humanoid cats in D&D in the first and second edition; they were later rewritten in the Red Steel campaign setting, and like Lupins, received a Dragon Magazine article (issue #247 expanding them into a multifarious "breed-based" racial class, allowing for the playing of every kind of catfolk from common "moggies" to full-blown sabertoothed tiger-folk and cave lion-folk. Their mystical origin, at least in the aforementioned dragon issue, is that they were created at the dawn of time when a female cat, the familiar of a human shaman, fell so deeply in love with her human master that an Immortal took pity on her and turned her into a human so she could marry him. Then the Immortal decided to see if she had fully acclimatized to being human and sent a mouse to run across the cave's floor; when she tried to catch and eat it, he was disappointed and turned her and her husband, who refused to abandon her, both into the first Rakasta.
In 3rd edition, when Mystara was lost in WoTC's files, they created a more generic anthro cat race, and they called them... catfolk. First appearing in the Miniatures Handbook, they were later reprinted with a fuller racial writeup in the Races of the Wild. They're described as basically a primitive and nomadic tribal society of humanoid big cats, most commonly resembling lions, with leopard, tiger and cheetah-like individuals being just a difference in coat patterns. They're described as having an emotional nature and tending to act in "fits and starts" rather than in a smooth, continuous effort. Bursts of activity interspersing consistent laziness. They have the habit of leaping impulsively out of hiding and into combat when a foe is in the vicinity, and this is among the reasons why they tend towards Chaotic Neutral alignment. Many catfolk favor the use of charms and totems that they braid into their hair for luck in battle, success on the hunt, and good fortune in other such endeavors.
Pathfinder also includes catfolk, which refer to themselves in-universe as the Amurran, as part of their fantasy kitchen sink approach. This elicited some query from fans over differences in artwork, portraying them various as catgirls and as catfolk, until it was clarified in one splat-book that Amurran are very "mutable" and so the exact ratio of cat to human in their appearance varies wildly between areas, or even between individuals, sort of like the Khajiit of The Elder Scrolls. Whether or not this has to do with humans tending to screw Amurran is left unclear.
Warhammer 40000
The back of the Warhammer 40k 6th Edition rulebook mentions "Homo sapiens hirsutus", or Felinids as one of the fifteen known races of abhumans. /tg/ generally likes to envision this as an entire race of catgirls, but accepts they probably look more like catfolk.
Red Dwarf
The first non-human "party member" of the Red Dwarf TV show was Cat, the survivor of a race of humanoids who evolved from Lister's pet cat over the three million years that the ship was flying off aimlessly into deep space. So, naturally, in Red Dwarf: The Roleplaying Game, Evolved Cats are one of the player options.
Unlike most catfolk, Red Dwarf catfolk aren't furries; they look like humans with slightly pointy ears and elongated canines. Sort of like downplayed elves with vampire teeth, really. Though according to dialogue in the show, they have six nipples, and it's implied the women thusly have six tits. They are characterized for being vain, shallow, short-sighted and self-centered, more interested in themselves and in looking as sexy as possible than in anything else, to the point of tending to ignore bigger issues to focus on their personal grooming.
The Elder Scrolls
Yeah, The Elder Scrolls is technically /v/ rather than /tg/, but it's popular enough on /tg/ to get its own article, so it counts for here. Skyrim's race of Catfolk are called the Khajiit, who're a race of magical felines whose bodies are affected by the phases of the moon when they're born. This results in khajiit being born as tiny, non-morphic kittens who variously grow up into humanoid cats, ordinary house cats, or giant cats that the humanoids use as battle-steeds. Yes, Nirn is a fucking weird place. They are most known in-universe for being expert merchants, specializing in a substance they create called "Moon Sugar", which is basically crystalized moonlight that you ingest like magical cocaine. Needless to say, they worship a lunar god-king called the Mane, which is only present in the sky when two other, lesser moons overlap.