You will often see the word written or hear it said especially around dojo’s it’s the Japanese equivalent of ‘innit’ … almost. Without going too deep it is used as a form of respect, a means off agreeing, saying yes and also ok. It is said when entering the dojo (training hall) and leaving both times with a bow. In fact instructors like to hear it not out of respect but because they need to know you are listening, understanding and accepting.
When used before kumite I like to think of it as you accepting the challenge of your opponent and thanking him for the use of his skills to test your own. It also means if there is any injury however slight it was done in the heat of the
moment without malice nor intent.
Finally we use it whenever we bow during class (except in kata). As Master Funakoshi said in one of his precepts, everything begins and ends with a “rei” or bow (and an oss). In other words everything we do in our life begins and ends in “respect”.