In fantasy games, pets can be gained from any of numerous different sizes and species. For most characters, however, pets will typically be derived from any of the common domesticated species, or those that can be domesticated, that live in the area where your character grew up. For magic users, maybe the pet is actually a magically conjured creature that appears or obeys only specific magical commands. Or maybe, your character wasn't able to find a pet, so they used polymorph or wild shape to become their own pet like a certain druid of Critical Role fame.
For sci-fi and horror games, pets may or may not be either living or mortal, and sometimes it could be both. Maybe your character's pet is a robotic approximation of a panther and frequently joins you in combat scenarios deep in the jungles of the alien world you are trying to conquer. Or maybe your character has a pet Cthulhu and its sole job as your pet is to destroy the sanity of any living creature you come up against.
Regardless of your game's genre or mythology, pets can add a lot of unique scenarios to a game. Especially when you gain the ability (typically from leveling up or learning a spell) to speak with animals who don't speak any of your character's known languages. It gets even more fun when your DM/GM roleplays your pet's side of the conversation.
Unless your pet is as un-useful as a goldfish, you will probably need to keep track of your pet's stats and abilities just like your would for your character. Even with the extra work, pets can be so much fun, even if your character transforms into their pet . . . yes, Keyleth (Critical Role, Campaign 1. Portrayed by Marisha Ray), we're looking at you.