Hog Rings are the metal wires wrapped around the seat support wires/frame and the through the seat cover. For reattaching the covers there are "Hog Ring Pliers", They look similar to regular pliers except the jaws have grooves in them into which the hog rings seat. This allows you to grip new hog rings and install them into the seat assemby. You should be able to get replacement rings and pliers at an auto parts store.
They got this name from actual hog rings. The real hog rings are larger and they are put into the nose (OUCH!) of pigs to discourage them from rooting with their nose.