Your pet is the character that you play as in Happy Pet Story, and they can be customized to your liking. Visit Barnie's salon, tap on her, and select "I'm ready for a makeover!" to customize your pet's physical features. Visit Tira's clothes shop to buy clothing for your pet.
Another feature you can customize is your pet's greeting. This can be done by tapping your pet's icon at the bottom of the screen.