Your familiar can carry one piece of equipment at a time, and will keep this even when placed back in its terrarium. Most familiar equipment can be equipped by only a single familiar, but a few pieces of equipment can be equipped by all familiars (except for the Comma Chameleon). Most of the familiar-specific equipment adds the equivalent of five pounds to your familiar's weight, but some have different effects. The Mad Hatrack is a special case; having no specific familiar equipment, it instead employs hats. The Comma Chameleon is also special in that it only takes familiar-specific equipment, that it destroys equipment you give it, and turns into that familiar for around 45 turns, and it always gets a weight bonus of +5. For both familiars, check out their pages for more information.
Most of the equipment specific to a single familiar is available only as a prize from the Cake-Shaped Arena. Other items can be obtained as noted.