There aren't too many differences between a nice hunter horse or nice hunter pony. A hunter in general needs to be able to to move well (long low floaty movement) with a happy and obedient disposition with correct form over fences with a nice big stride. A competitive horse should be able to do it's lead changes and go down the lines straight and smoothly without hesitating at the fences. A pony is no different. Technically a pony should be able to compete in most classes that a horse will, but when you get to the A rated shows there are special divisions for just ponies. In some of those divisions you're going to need to "model" which is basically a look at the pony's confirmation, or "jog" after a jumping class, which is a trot in hand in a bridle but no saddle checking for soundness. So if you're looking to show in rated shows you need something that will model well. The sizes go as follows:
Smalls: up to 12.2
In the regular pony hunters they jump 2"3, 2"6, 3" respectively (this is different for the greens)
Some people avoid the smaller horses of their sizes (like a 12.3 medium pony is sometimes nicknamed a 'small medium') because 'small medium' will generally have a harder time making it down the lines in the correct number of strides as say a 'large medium'. All sorts of great info can be found on www.usef.org
under rules and governance.
On top of all of the show details you of course want the same typical stuff: nice obedient friendly personality that a child should be able to ride safely etc.