The Pets aspect to Pet Rescue Saga is that on the majority of the levels there are lots of animals that you have to rescue from what the game calls the Evil Pet Snatchers. You can see pets that are in need of rescuing sitting above the blocks - you have to clear a path to the bottom in order to free them, by bursting blocks.
Once the pets come into play they become a key part of completing or passing each level, often requiring a certain number of pets to be saved before you are permitted to move onto the next level. Should you get near the bottom of the board and discover that there is no way for you to save the pets required, the level will be failed.
Having to save pets adds a new dimension to the game and, as the name of the game itself suggests, should be your initial focus on each level - they are where the big points come from.
Save those Pets!!!