You can obtain this pet by opening your gift box item (that you receive in the mail) and accepting the quest Welcome! which requires you to turn in the gift item to an NPC in exchange for a choice of three pet items, one of which is the Diablo Stone. The name of the gift box, NPC and quest zone is different depending on your chosen race.
Mini Diablo is inspired by the title villain of Blizzard's Diablo series of games (the name is literal; it's a very tiny version of the Lord of Terror). As its idle animation, it will occasionally breathe fire.