Companions are small creatures who follow the player around. They have no function and provide no benefit, aside from following the player.
To enable/disable companions select 'Companions' inside the 'Wardrobe' option in the inventory menu.
Some companions will be available as special promotions - either through events or as rewards for limited events.
Every unique small animal (i.e. Critters) in the world can also be collected as a companion. Each critter must be fed its favorite or liked food at least twice (frequently more) to register it in the Collection. Once registered, that critter will be available as a companion inside the 'Wardrobe' option in the inventory menu.
Since only the first successful feeding per day counts towards collecting a critter, recruitment will take multiple real world days.