"Koryu" (古流) literally translates to "old school" or "old style" and this refers to Japanese martial arts schools founded prior to the Meiji Restoration in 1868.
The Meiji era marked a period of significant social, political, and cultural changes in Japan, including the modernization and westernization of various aspects of society. With this there coincided a sharp decline in the practice of kobujutsu and the number of surviving koryu. of course many koryu had gone extinct previously either with a lack of practioners to pass on the art or with ryuha developing and changing into new ryuha. Koryu are inherently historical and classical having their origins in Samurai traditions and methods of combat and self defense. (note, however, that there are non-samurai martial arts ryuha as well).
We should address martial arts which are not considered koryu. Usually we refer to martial arts founded after 1868 as "gendai budo" (現代武道), meaning "modern martial arts". Gendai budo typically include traditional elements, with modern elements (methods, context, philosophy, etc.). The clearest and most familiar examples would be Judo, Aikido, and Kendo. Koryu is not "better" or "worse" than Gendai Budo in any way, it it is important to point out that there is a distinction. I'll oversimplify and say that the distinction is historical context and development.
There is a spectrum among koryu between specialized schools and comprehensive schools (sogo bujutsu). That is, some schools include as large number of weapons, methods, and strategies of armed and unarmed combat and other traditions not directly related to combat. The, there are some schools which are very specific to what they teach. Of course there is some in-between. Yagyu Shingan Ryu is one of those comprehensive arts which includes methods of Yawara (Jujutsu) both armoured and unarmoured, kenjutsu, battojutsu (iaijutsu), bojutsu (staff), Naginatajutsu, Sojutsu( spear), Torite (Arresting/Capturing), Kappo (Healing/Resuscitation) and more. Kurama Ryu is a school of kenjutsu only, specifically odachi, and relatively few techniques.
It is fairly common to find organizations which practice several of these specific styles in order to have a more comprehensive system of practice. As well it is fairly common to find elements of these sogo bujutsu koryu to exists in disparately among several organizations.
What is essential in koryu is the student-teacher relationship and transmission. Training in koryu is a serious endeavor. Students and and teachers practice and teach diligently in order for the koryu to be transmitted accurately to the next generations. Koryu groups tend to be small with a lot of direct contact between teacher and student and hands-on instruction (literally). While public demonstrations of the style is common, the true meaning of techniques, principles, and underlying philosophy are communicated verbally by the teacher to the student as part of the oral tradition and transmission. This creates a direct and personal connection between generations of practitioners. Beyond physical techniques, Koryu teachers pass on the cultural, ethical, and philosophical aspects of the art. This holistic approach ensures that students not only become proficient martial artists but also understand the broader context of their training.
The student-teacher relationship in Koryu Bujutsu is a dynamic and multifaceted connection that transcends the classroom or dojo. It reflects the traditional values of respect, loyalty, and the responsibility of both parties to preserve the legacy and authenticity of the martial art for future generations.
Further reading:
A good article on this can be found at koryu.com website: "A Koryu Primer" by Diane Skoss