In a HALTER class, it's all about conformation. You can touch your horse and everything, because they are just being judged on the horse that shows the best confo/type for that breed. Almost every breed has a halter class, and I'm not sure on what other breeds require, but for stock horse types, it's about being like.. a quarter horse type. Stocky, compact, straight... That goes for appys, paints, and quarter horses. They have to be able to stand square.
For SHOWMANSHIP, it's judged on how well you show the horse. Confo counts for some, in certain shows, but not a lot. the scoring usually goes like this: The appearance of the horse is 40 points: 15 for condition and thriftiness, 20 for grooming, and 5 for tack. Showing the horse is 50 points: Ring deportment is 10, actions are 10, leading is 15, and posing is 15. and last, the exhibitor is 10 points: 5 for appearance, and 5 for attitude.
Basically, you're looking for a horse with the best possible confo, a lot of eye appeal, willingness, and trainablity, IMHO.