What is a schoolmaster?

A schoolmaster is a well trained, older horse, generally one that has competed at a higher level and is now semi-retired, that is suitable to use to teach riders the finer aspects of an equestrian sport. So, you might talk about a jumping schoolmaster, reining schoolmaster, dressage schoolmaster, etc.

A schoolmaster may or may not also be usable by beginners.

Note that the shorter form "schoolie" refers to any horse being used regularly in lessons, and is a short form of "school horse" not "schoolmaster" - which is much more specific.