Here is the full list of all 11 Desktop Pet achievements.
Own 2 pets.
Change the eyes of a pet.
Change the mouth of a pet.
Change the head of a pet.
Change the tail of a pet.
Equip glasses.
Equip a hat.
Sell a potion in the Potion Party mini game.
Use a name tag.
Change the color of a pet.
Buy a toy.