Curb

A curb is a form of ligament damage that affects the ligament just below the hock, at the back of the leg. With a "real" or "true" curb, there is swelling as a result of the body laying down fibrous tissue in that area. Because the tissue is permanent, the swelling is generally permanent as well, but it's usually not painful after the acute phase, nor should it interfere with the horse's way of going - especially if it's an old curb (not a recent injury) and the horse is going sound. The treatment for a recently-acquired curb would be the same as for any other damage to a tendon or ligament

A "false curb" is something else entirely - there's no soft-tissue swelling, fibrous or otherwise; it's nothing more than an enlargement of the head of the splint bone (and this is not due to injury - it's conformational and, more often than not, hereditary). Some vets see this as a sign of weakness, but again, if the enlargement isn't recent and the horse isn't a youngster, it's generally not a problem - and isn't even considered to be particularly abnormal.