"Fruits Basket" is the story of Honda Tohru, a young girl whose mother is killed in an accident. She goes to her paternal grandfather for help, but isn't really accepted. So she camps out in a tent on the Sohma family land, eventually being discovered by Sohma Yuki and Sohma Shigure, cousins who live there.
They convince Tohru to stay at their house instead of camping out, and while staying with them a young, red-headed manc rashed through the roof and proceeds to attack Yuki. This is Sohma Kyou. Tohru, trying to stop him from attacking Yuki, trips and catched Kyou around the waist.
*Poof!* Kyou turns into a cat, the reject of the Chinese Zodiac. Freaking out and fearing that she did something wrong and somehow turned Kyou into a cat, Tohru manages to turn Yuki into a rat and Shigure into a dog. When she calms, Shigure explains that they are cursed.
The Sohma family takes the form of the Chinese Zodiac animals when either extremely nervous, or when hugged by a member of the opposite sex who is not afflicted with the curse. Kyou, being the cat, is even looked down upon among the cursed, because the rat tricked the cat into not going to the feast the other 12 animals went to, and so the cat is not part of the zodiac. ThereforeKyou transfers all his frustration into being unable to beat Yuki, the rat, who he feels embodies his enemies.
The other family members, including a few girls, also are cursed. Sohma Kagura, for example, is the boar. Sohma Hiro is the sheep, Sohma Momiji is the rabbit, Sohma Kisa is the tiger, Sohma Hatsuharu is the ox (he actually turns into an adorable black and white moo cow!), and Sohma Hatori is the dragon (he turns into a seahorse).
Momiji after he hits puberty...kinda looks like Yuki with blonde hair ^_^
The rooster and the horse are missing, although I've heard that Sohma Rin, Hatsuharu's ex girlfriend, might be the horse, and that Tohru's friend Uo-chan has a Sohma boyfriend who just might be the rooster (manga only). There is also a Sohma named Kazuma who is the martial arts expert that trained all the cousins. He took Kyou in to redeem the memory of his grandfather, who was also the cat.