As you complete a certain amount of levels in the game, you will have the chance to find a new pet.
To complete the rescue, you must visit some neighbors and answer questions of varying difficulty and category. This way, they will agree to give you more information about the location of the pet you are looking for.
Once you find it, you will be given tasks to feed it, cure it, bathe it and give it a toy. You will also have the opportunity to decorate the pet's living space in the shelter.