Pet vendor
Pets are unlocked after speaking to the leader of the Markans. A merchant appears in the rebuilt center-most building of the town. He sells the 'Summoner,' Pet Place,' and 'Pet Food.'  The Pet Place must be placed on a floor tile.  To feed your pets you left click on the food in your inventory and then left click on the food bowl in the Pet Place. Pets gain experience from eating and will only eat when their hunger reaches 0.
Pet vendor inventory
There are four pets you can summon: Slime, Ice Pet, Robopet, and Rabbit. Each pet requires different items to summon and has their own special ability.

You can only have one pet active at any time. If you want to swap one out the previously summoned pet returns to your inventory as an item and keeps its experience and levels. Pets can walk through obstacles such as walls, trees, snowdrifts, etc. Additionally, pets cannot take damage and other creatures do not react to them.

Pet place with bunny