Japanese martial arts are usually defined in two groups. Post-modern and modern. There is no single agreed upon criteria for the separation. Generally, schools that were founded before 1868 are referred to as koryū 古流 and arts after that, gendai budo 現代武道. The year 1868 is when the samurai as the ruling class was beginning to be abolished and the Meiji government was being established. Some point to the Hatorei edict in 1878 which restricted the wearing of swords.
In Japan, the terms koryū and kobujutsu 古武術 are used interchangeably. Some people distinguish between the two, however, it might be a semantical issue. I would rather not argue that. Linguistically koryu and kobjutsu are different though.
Ryū 流 is also called nagare in Japanese. Nagare means to flow. In the classical martial arts of Japan, the flow is the information and techniques of the school from teacher to disciple. The character 流 depicts the information or secrets (inner thoughts) of someone in a river which carries this information into someone else. More precisely in bujutsu realm, the complete system was given to the someone with the certificate Menkyo Kaiden 免許皆伝 (complete transmission). This flow of information from one to another is what makes Koryū literally that which flows from what came before. Kobujutsu 古武術, however, just means old martial arts.
Self Defense & Techniques
The koryū and kobujutsu are not for self-defense. They are not sports either. The kobujutsu are traditions of fighting. While their goals might be combative in nature and focus less on personal development than say, modern kendo, they are not for street fighting, modern war, nor home defense. People often conflate martial arts and self-defense. It is my view that they are two distinct things. Towards the middle of the Edo period samurai and others that romanticized the past and the era of battlefield combat or became tired of the kata methods turned towards the use of the shinai and shinai-bukuro for a safe means of sparring and 'testing' things out. This spread to competition among dojo. You can certainly imagine the ego's behind this form of competition.
There are many stories of schools (ryū-ha 流派) trying to dominate local areas through the use of these inter-school challenges. It was the late Edo period that saw a rise in the challenges of other schools. This was called Taryūjiai 他流試合 where swordsman of different ryū-ha could compete with each other to see who and who's style was the best. These were not always friendly and just for fun affairs. It is interesting to have inter-school bouts for educational purposes but anyone bragging that they still engage in such activities for anything other than private education between schools is a little immature in my view. In general, there are no competitions in Koryū as exist in modern kendo, karate, or iaido.
The classical martial arts of Japan is a unique and interesting tradition of fighting methods and tactics from hundreds of years ago. Some schools that came later were considered watered down variants but in today's view, they are also viewed as koryū. The purpose and application have really been lost in time no matter how serious the teachers and students in the last few hundred years. Although to use an example of kobujutsu or koryū kenjutsu schools the idea of being effective is paramount as compared to something like modern iaido. Techniques of the sword should be based on realistic principles of fighting with the katana.
If someone leaves a ryū and starts up their own school, even using the same techniques or modified slightly then purists will argue it is not koryū. Although in many koryū there have been significant changes, those changes were implemented by a menkyo kaiden. It is therefore still a legitimate koryū by many of the purist's ideas. I am not convinced of this and my ideas are a little less rigid. It should also be noted that once you change things to a certain extent it loses its connection to the source or founder. Many schools in postmodern Japan emerged from other schools. There were hundreds of sword schools alone that branched off from such schools as Nen-ryū, Shinto-ryū, and Kage-ryū.
There many koryū still extant in varying degrees of completeness from their origins. Some schools have been lost due to untimely deaths or fires or earthquakes and floods. Other schools have lost part of their original techniques to such tragedies. Waza and tactics were often transmitted physically by person to person and not written down in great detail. This is known as kuden 口伝, transmission by mouth.