The term "dressage pony" is sometimes used interchangeably with "show pony". However, not all show ponies are trained in dressage (and in fact not all show ponies are trained in the English disciplines), and not all ponies trained in dressage are show ponies.
Thus it might be easiest to use an analogy with the bio-horse world: dressage ponies are simply human ponies that have some dressage training - whether those ponies choose to compete in shows is completely separate to how they are "schooled" or trained.
Dressage ponies, just like their bio-equine counterparts, are expected to perform additional advanced movements (depending on how far they have progressed in their training) in addition to the standard gaits (walk, trot, canter). Both in schooling and in competition, a pony is judged on the overall quality of movement and in addition to his "willingness" and submission.
Moreover, dressage ponies are often well groomed (e.g. nicely braided mane) and have fancy tack, and their rider or trainer will often be well turned out as well.
Human pony dressage competitions do exist (for example, the Los Angeles Pony and Critter Club often incorporates a dressgae test or two in their annual Jamboree), and their structure is similar to USDF shows (though typically human pony dressage tests are only given for one or two levels).
As with most things pony play, you can check out the resources for the bio-equine equivalent to get guidance for how to structure your pony's training. In the case of dressage, I would suggest taking a look at the United States Dressage Federation - you can even download practice tests from training through fourth level. These tests are great for fun, or preparing your pony for shows, and they're completely free.
While some ponies focus exclusively on dressage, many are cross trained in dressage along with another discipline. In the bio-equine world this is usually done for eventing horses. However, there are not too many human ponies that do eventing, thus most human ponies who are cross trained do so simply because the pony and/or trainer enjoys dressage.