The quarter method of showing is essentially a way to make sure that the judge always has an unobstructed view of the horse in the ring. When moving forward, the handler is always on the left side, as is "correct" in any situation (getting the horse accustomed to working from both sides is a good thing, too, in schooling in hand). When standing still, however, the horse is to be between the handler and the judge. They system divides the horse into quarters, with a line from muzzle to tail, and a line crossing the withers.
The handler never stands in the II or III quarters, but moves from I and IV as the judge inspects the horse. Every time the judge crosses an imaginary line, the handler switches sides. For example: The horse is standing square, and the judge begins to circle the horse and inspect the grooming job. He is in I, and you (the handler) are in IV. When the judge crosses the wither line, into III, you step from IV into I. When the judge passes the tail line, you step back from I into IV. When the judge crosses from III into IV, you go back to I. When the judge crosses directly in front of the horse, you cross in front of him back to IV.
Here's a video to help explain, and to show the footwork involved to make the switches look fluid. Also, note that in the beginning of the video, before the pattern starts, the handler sees that the judge would be in the IV quarter if he were closer, and so switches to I until the pattern starts. This is very correct, and should also be done once the individual entry has been dismissed to the lineup. Disclaimer: Not me, not my horse in the video.
I hope that clears some stuff up about one of the more confusing parts of G&S. As I said above, many halter exhibitors do this as well, not so much out of the requirement, as in showmanship, but to make sure that the judge's view and ability to see the conformation of the horse is unobstructed.