As you progress through the game, you will have the chance to find new pets.
To complete the rescue, you need to visit some neighbors and answer questions of varying difficulty and category. By doing so, they will agree to give you more information about where they saw the pet you are looking for.
Once you find it, you can place it in the shelter and decorate its space. You can also feed it, cure it, bathe it or give it a toy every time a cloud of thought appears over its head.