You need three things in order for your pets to breed. A male and a female cat or dog and a doghouse. Sims can direct a pair to breed by telling one of them to try to make a puppy/kitten with the other. If their relationship score is high enough (more than 25 points daily), they will go to the doghouse and woohoo in typical Sim fashion.
If the woohoo is successful, a litter will follow a couple of days later. Litters can have up to 4 babies, depending on how many pets are already in the house (6 is the max).