The –Ryu
All koryu and several gendai budo schools are called such-and-such-Ryu. The -Ryu appended to a title refers to the system's "name"; it is a "style" or "school" delineation. Therefore, there karate do, while a unified gendai budo fighting system originating out of Okinawa, retains several strains that are quite different in their kata and training methods. A Shotokan stylist would be different from, say, a Goju-Ryu stylist, as opposed to an Uechi-Ryu stylist. They would all be doing karate do, certainly, but even the application of a simple punch would be very different in terms of placement of the arms, use of force, and so on.
Some gendai budo do not have any -Ryu systems. Kendo, for example, has been unified as a national (and international) sport and pastime for the use of bamboo staves to represent sword fighting. There are no "styles" or schools of kendo. There is only kendo. All kendo schools share the same general rules when it comes to competition and kendo kata (forms). Likewise judo, although it has fragmented a lot in recent times, is basically judo. Even recent iterations of some modern "jujutsu" schools are, at heart, simply subsets of some parts of judo, emphasizing perhaps the more combative or roughhouse aspects of judo to the detriment of the sportive aspects.
Aikido started out as one and only one martial art form. It, too, has developed along different stylistic lines due to differences of style, personality clashes, and other kinds of martial arts politics. Aikido arts presently encompass various different schools, the larger ones being Aikikai (or hombu, the "main branch" style of aikido), Ki Society (Ki No Kenkyukai), Yoshinkan, and Tomiki-ryu.
The differences between koryu schools are even wider. While a Shito-ryu stylist may find similarities in a kata performed by a Shotokan stylist, some jujutsu schools, for example, have no common grounds with each other. Even similar techniques may go by completely different names.
Koryu Schools
The number of distinct koryu schools have decreased since Japan's modern era (1868), due to Westernization, the consolidation of martial endeavors into the modern -do schools, and sheer neglect. However, a number of koryu schools continue to this day. While koryu budo once numbered in the thousands, they now number in the low hundreds, if at all that much. The list that follows is very incomplete, but suggests the variety of possibilities still available in koryu schools. Some of the schools may not fit in the category I have assigned them, depending on one's point of view. For that, I beg your indulgence as I admit to not having an in-depth experience with most of the schools listed.
Composite Schools (primarily a variety of bujutsu methods)
Tenshin Shoden Katori Shinto-Ryu
Kashima Shin-Ryu
Kashima Shinto-Ryu (Bokuden-ryu)
Takeuchi-Ryu (Honke, bunke, and Bitchu-den)
Jujutsu
Daito-Ryu
Araki-Ryu
Shito-ryu
Tenshin Shinyo-ryu
Sosuishitsu-ryu
Takagi Hontai Yoshin-ryu
Kito-ryu
Iaijutsu
Muso Jikiden Eishin-ryu
Muso Shinden-ryu
Hoki-ryu
Mugai-ryu
Sekiguchi-ryu iai
Hayashizaki Muso-ryu
Tamiya-ryu
Mugai-ryu
Hasegawa Eishin-ryu
Swordsmanship
Yagyu Shinkage-ryu
Ono-ha Itto-ryu
Sekiguchi-haItto-ryu
Hokushin Itto-ryu
Hyoho Niten Ichi-ryu
Jikishinkage-ryu
Maniwa Nen-ryu
Shizen-ryu
Staff (bojutsu, jojutsu)
Chikubushima-ryu
Shinto (or Shindo) Muso-ryu
Kukishin-ryu bo
Shuriken jutsu
Shirai-ryu
Negishi-ryu
Sojutsu
Owari Kan-ryu
Hozoin-ryu
Kyujutsu
Ogasawara-ryu (also a school of etiquette and horsemanship)
Takeda-ryu
Naginatajutsu
Tendo-ryu
Jikishinkage-ryu naginata
Kusarigama
Jikiyuishin-ryu