If you place first or second in your halter class, you go in for the junior or senior grand (according to your horses age). Then if you place first or second there, you are eligible for the Grand class. You must show in the grand classes to keep your points!
Supreme is a non rated (just for fun and bragging rights!) class for all horses (mare, gelding, and stallions) to show against each other for the title.
I got busted big time for saying this one wrong. To get into the Championship class, you must place first or second in your age/height class. In AMHR you have Jr Champion and Sr Champion, not Grand Champion yet (this one got me a slap on the wrist for sure). Then the Jr Champion and Reserve and the Sr Champion and Reserver go back into the Grand Champion class. This is for each division, stallions, mares and geldings. Once you have a Grand Champion Stallion, Mare and Gelding, those go back into the Champion Of Champions class (AMHA has the Supreme, same thing, different name). One horse is picked as the Champion Of Champions. If you have more than one judge, each one will have their own set of winners.
Since AMHA does not show their Jr horses against their Sr horses for a Grand Championship, the winners in those Championship classes are called Grand Champions, ie.. Grand Champion Jr Gelding. For their Supreme class, you will have six horses. The Grand Champion Jr horses and the Grand CHampion Sr horses.