Instructions
Create two animals when you create your family. One must be male and the other must be female. Dogs and cats cannot breed with one another. The animals must also be of similar size. A large dog cannot breed with a small dog, for example.
Purchase pets in-game when you visit the pet store in your Sim's neighborhood. Stray pets will visit your Sim's home. Click on the stray and interact with it to build a high relationship with your stray. You will eventually have the option to "adopt."
Select one of the pets and then click on the other pet. They will have social interactions. Click on these interactions and repeat this process to build a relationship between the two animals.
Purchase a pet house in "Buy" mode and place it somewhere on your property. Select one of the animals again and then click on the pet house. Select the "Try for Puppies/Kittens with..." option and click on the name of the other animal when it appears.
Turn on your computer speakers. After the animals use the pet house, you will hear a lullaby. If you do not hear the lullaby, repeat Step 4 until you hear it. When the puppies or kittens arrive, you can visit the pet store to sell them if you choose.