Your cat in Harvest Moon DS Cute is an animal there purely for companionship. It doesn't need to be fed, and you can't breed it.

You can build up your cat's affection for you by simply picking it up every day or by petting it with the touch panel glove.

Your cat can be a useful tool for raising people's LP/FP points, from the showing your pet trick. Other than that, your cat doesnt do anything benefitial.