The first and most common method is a traveling circle, which involves moving with your horse as the horse goes in a circle around you, the person's feet follow a small circular path within the horse's larger circle. Being a very dynamic, driving way of circling your horse, it establishes dominance very quickly and is particularly useful in schooling defiant pushy horses. A traveling circle is also very beneficial in teaching a horse to match your energy and translates directly to having a horse seek unity with you when you are riding.
The second method is a stationary circle, where the person stays still allowing the horse to travel around them while passing the rope behind their back. This method is less used, harder for humans to learn, and seems to be misunderstood by many trainers, so I will focus on the many benefits of this technique.
Staying still while your horse does all the moving teaches your horse to be responsible for managing his own emotions and energy, he has to rate himself without being constantly driven forward, which is what we would all like our horses to do when we are riding. It is also easier for a person to be more consistent with their cues on the long line. A moving person has trouble keeping the same amount of distance from their horse so it is harder for the horse to learn to maintain slack. When the human is still, the establishing an even distance between horse and human becomes the horse's responsibility, and the horse will learn to travel a perfect circle with slack in the line much more quickly.
A stationary circle is also a fabulous diagnostic tool giving you real insight into what your horse is thinking. Constantly turning or moving with the horse can be very intimidating to very shy sensitive horses and they won't give you the feedback you need to really understand what they are feeling. I have heard people say that staying stationary allows the horse to "cheat" the person and slow down or stop behind the person's back. I disagree with this. A few lazy horses will stop behind you to avoid work, but if a horse stops behind you, most likely the horse is intimidated by your gaze. Constantly looking directly at your horse feels like predatory behaviour and your horse wants to stay behind you where you can't stare at them.
Recently, I have had the opportunity to play with one of the Jicarilla mustangs who has been in her adoptive home for about a year. She had been taught to circle in a round pen with the continually driving method, which taught her to be very attentive to people and their energy, but when I tried putting her into a trotting circle without the round pen to support her, I got a chance to brush up on some of the sand skiing techniques I haven't used in a while! By trying to stay as still as a fence post in the center of her circle she learned very quickly to leave slack in the rope, run behind me and stop. I understood she was simply seeking comfort, so I just repeated sending her on the circle again and again until she finally made it all the way around to my front side again, then I invited her in to me for a snuggle. By doing this she eventually learned to make one whole circle and come back for a pet. From there I was able to ask her to keep going for a second circle before coming back, now she will continue making relaxed trotting circles with slack in the line until I ask her to come back, but each time she passes in front of me, rather than being worried or intimidated, she looks at me as if to ask, "Can I come back this time?"
She still tells me she would rather be with the other horses than with me because she slows down some and draws out wider on the circle when she is nearer to the other horses. This is the sort of feedback you will never get using a continually driving circle, because they do not feel free to stop or slow when they want to. Knowing how your horse really feels is the first step to addressing the things they are afraid of, or to help them gain a better opinion of people.