(thanks to Diana for the question)
Many people have heard "Osu!" in various martial arts schools. Some students even say "Osu!" without really understanding what it is about. Here's how "Osu!" is used:
"Osu!" is generally said to any higher ranking belt (especially dan ranks and instructors) whenever meeting them.
"Osu!" is the response to any order or feedback given. It is the response a student gives whenever any instructor announces a technique or the start of a drill, or anytime an instructor corrects a student's technique.
3) An expression of commitment (Kiai)
"Osu!" is used anytime you count out a drill for the last rep. It is also an expression of concentrated intention in the way a kiai is done in other martial arts.
OK, fine. But why "Osu!" and not "Hai-Ya!" (or any other random sound)?
Traditional Martial Arts and Militarism
Many of the traditional arts (especially aikido) have very close ties to militaristic expression in Japan. It is customary to greet senior soldiers (NCO and officers) with "Osu!" much in the way it is used in Yoshinkan aikido. This may be a shortened form of "Ohayo Gozaimasu" which is now extended to be an acknowledgment and compliance with whatever the speaker has said. Much of the post-war martial arts training in Japan is very closely linked to right-wing (and, indirectly, to support of the Emperor). Yoshinkan itself is closely linked to the Tokyo Metro Police Force, which is also a very right-wing organization.
"Osu" is also said to be comprised of the character for push "osu" and the character for perseverance "nin". This is a reminder to train hard, and early Yoshinkan disciples trained very hard indeed. We honor their spirit and commitment to training whenever we say "osu!". It recognizes our intention to train just as hard as they did, and to follow their example of "shugyo".
More to add? Let me know.
See you in class.