Some mean boss tomcats are guarding the better neighborhoods of Whiskertown and to move up in the world (or the alleyways) you need to defeat them. To be eligible to fight the boss you need to wake him up first by successfully pouncing on the same number of cats as indicated on the boss icon.
To defeat the boss, make sure to launch your cat at them by tapping on it when the moving slider on the right is in the green area. Launching a cat while the slider is in green will result in a strong attack.