This is the story that Kyoko tells Tohru when she is a little girl. Tohru felt so bad for the cat being left out that she decided that she wanted to be born in the year of the cat rather than the dog. When discovers that the Sohma family is possessed by the spirits of the animals from the story, she's ecstatic to hear that the cat is represented. When she meets Kyo, she tells him that she always wanted to be a cat, which needless to say, catches him off guard.
As time passed, the man decided that maybe he would be able to make friends with other animals. He wrote many invitations for a grand banquet and sent them out to all the animals. Twelve showed up and together they would have a wonderful banquet every night under the moon with god and the cat.
Eventually, the cat became ill and god became devastated at the thought of losing him. Together, he and the other animals cried and realised that their banquet would eventually end because they all must die at some point. God had an idea: he took a bowl of water and drew his finger in a circle upon the water's surface. He made the cat take a sip and told the other animals that if they also drink from the bowl, they will be reborn time after time and they would all be able to have a banquet together for eternity. The animals willingly drank from the bowl but the cat asked god why he was made to drink from it: he did not want to live for eternity.
The cat told god that it was okay that he was afraid but he and the others must accept that everything must come to an end, even if it is very sad. He tells god that even if it was but for a short time, he was happy every day they spent together. The cat says that if he were to be reborn, he would rather see god smiling in the sunlight, surrounded by many people, laughing together. Those were his last words before he peacefully passed away.
God was happy but the other animals no longer cared for the cat as they did not understand why he would reject eternity together. They continued their banquets until each animal died and then god himself died. They were reborn time and time again until the origin of their meeting was all but forgotten until they are released from the curse many, many years later.
The cat has become warped over the centuries of rebirths, not just in form but in representation as well. Akito says that the cat's duty is to be the worst of the lot so that the other accursed members can feel better about themselves… and this is what has happened as a result. Even various members of the zodiac admit to feeling better around Kyo because no matter how horrible their lives might be, he has it worse.
In the Vietnamese zodiac however, the cat actually does have a place in the zodiac! It's the fourth animal, replacing what would be the rabbit in the Chinese zodiac.