The story of the Cat and the Rat’s rivalry is very popular. There are multiple versions. The one I will tell you I watched on an anime called Fruits Basket.
A long time ago, the Cat and the Rat were good friends and neighbors. The Jade Emperor had told all the animals that he was holding a banquet the next day. The Rat told the Cat that the banquet was the day after next. The next day, the Rat rode up on the back of the Ox. When the Ox arrived at the banquet first, the Rat jumped off the Ox and was declared the first in the Chinese Zodiac system. Meanwhile, the Cat was at home thinking tomorrow was the banquet. When he discovered the Rats trickery, he swore to always hate the Rat and cats still hunt the Rat today.
For other versions of the story of the Cat and the Rat, try these other sights: