Like Myboypuck said normally they stick you on the most dead broke horse they got and see what you know.
Here is the list of what I think each level should be at in order to be so called started to be labeled higher. This is only for riding though... when it comes to jumping and other special stuff it changes a lot. The same for ground work stuff which I think should be accomplished as soon as possible before getting into the saddle.
Beginner - Needs to know how to groom, basic saddle and tack knowledge, basic horse knowledge on breeds.. health and such. Know what the gaits are, and how to stop (also emergency one rein stop).
Novice - Knows how to walk, maybe a little trotting but not by themselves.
Advanced - Knows how to mount and dismount, knows how to walk and turn in a controlled manner. Knows a little how to side pass and forehand turns. Knows how to post at a trot. And a little on how to back.
Intermediate - Knows how to trot and turn in a controlled manner. Knows how to canter in a single direction around the ring (clock wise and counter clock wise... but doesn't need to know lead changes and how to turn the different direction within the ring at a canter). Knows how to do forehand turns, haunch turns, side passes.
Independent - knows how to do all the basic riding skills all by themselves with obtaining knowledge of others without much help. They can explore and find many results out of trying certain stuff.
Expert - has very good knowledge and experience on every aspect of riding to the point of teaching others.