The sword arts themselves are divided several ways. First is by type, ken or iai (sometimes called batto). Also at the same time they divide by origin, the three families of sword arts; Muso ("empty"), Kage ("shadow"), and Shinto ("new sword") styles. Some types mix these dependant on origin and application. Within each type are the three styles. All can be recognized by their employment of various charateristics.
There are two organizations within Japan that have tried to consolidate these sword arts. The Zen Nihon Kendo Renmei (All Japan Kendo Federation) oversees certification requirements for kendo, kenjutsu and some iaido ryu. The Zen Nihon Iaido Renmei (All Japan Iaido Federation) is a much younger organization and oversees certification of some kenjutsu and many iaido ryu. However, the more traditional of the ryu have not joined either of the two organizing bodies and have remained independent and free of oversight.
It should be finally noted that many martial systems included sword arts as a part of their curriculum. In other cases, predomanently sword martial systems have included other types of weapons. And finally there are still other systems which teach only sword arts. The most traditional ryu would teach the sword as their primary weapon, and perhaps some other arts (weapons or empty hand). Only in modern budo or bujutsu have the empty hand arts become primary with the sword playing a supporting role.