Most pets are only "alive" for 90 days, although sometimes permanent pets are available in the Cash Shop.
After 90 days, your pet turns back into a doll, and it needs to be revived before you can play with it again. To revive your pet, purchase a Water of Life from the "Pet" → "Pet Use" category of the Cash Shop.
You may also be able to find it in the Maple Rewards Shop (the "Special Promotions" → "Maple Rewards Shop" category of the Cash Shop. Double-click to move the item from your Cash inventory to your character's inventory. Then double-click the item to revive your pet.
If you have more than one pet, you can select which one you want to revive. Reviving your pet will give it another 90 days of life.