If your sims want their pets to have their own children and you don't know how to do it, you are at the correct place. There are some several things that you should do in the game:

    1.  Have the pets have high relationship. 
    2.  Have the pets at the same lot. 
    3.  Have the pet and the owner have a high relationship. 
    4.  If you are breeding cats or dogs, you have to have a dog/cat house (can be found in Buy Mode)
    5.  If you are breeding horses, you have to have an Equestrian Center in your town. 
    6.  To breed horses, you don't need to have two horses with high relationships. You can just take them to the Equestrian Center and select "Breed...".

Breeding Cats

To breed cats, you have to have two cats with high relationship, a sim and a cat that have high relationship and a dog/cat house in your lot. Both of the cats and the owner should be at the same lot. When you have the right condition, your pets should try for baby. In order to do that, when your Sim is active, click on the pet and select "Try for baby". Another way to do it is to choose "Try for baby" when you cat is active.

Breeding Dogs

To breed dogs, you have to have two dogs with high relationship, a sim and a dog that have high relationship and a dog/cat house in your lot. Both of the dog and the owner should be at the same lot. When you have the right condition, your pets should try for baby. In order to do that, when your Sim is active, click on the pet and select "Try for baby". Another way to do it is to choose "Try for baby" when you dog is active.

Breeding Horses

Under construction.

Please note that these steps are not accurate as I wrote them from my mind. I will edit the information as soon as I can.