Just an orthographical note.
"Osu" as a greeting (contracted from "Ohayossu", contracted from "Ohayou gozaimasu"), is written in hiragana おっす, and as Yawata-san suggested, can vary by region. It's like "Hiya" or "Howdy".
"Osu" as a response in the dojo is written with ateji kanji (kanji with irregular readings) 押忍. It's essentially a different word from the other (although they probably came from the origin) in use and orthography. They look the same in romanization, but they are two different words.