"Natural Horsemanship" is a brand name / fad name / pick your favorite epithet.
What people generally mean when they say any of these names is "Good Horsemanship, as opposed to Bad Horsemanship, and in particular using a horses's natural mechanisms to teach and communicate with the horse instead of forcing the horse into trying to understand what you mean through blind trial-and-error until they eventually either get it or give up and ignore the world."
Here's a concrete, if extreme example:
Someone who is a practitioner of "Bad Horsemanship" might raise a foal in a stall without ready access to other horses except in rare, human-supervised interactions. Then as the foal gets older, when it does something like crowding the human or nipping at him, the owner (trying to be compassionate but not understanding) might lunge the colt around-and-around to "teach him not to do that" without actually striking the horse. The horse, not able to connect cause-and-effect between these events, just gets tired and thinks of the human as someone who wouldn't let him quit doing something when he was tired.
On the other hand, a practitioner of "Good Horsemanship" might find a herd situation to let the foal spend a couple of years in as it grows, so it can develop appropriate social skills among other horses. Then when the colt is older and does something like crowding the human or nipping at him, the human will be on the lookout for such behavior and arrange for the colt to "accidently" bonk his nose on the human's elbow right when he reaches out for the nip. After a time or two of this, the colt will realize that when he tries to nip, he bonks his nose, and will give up.
If the person's timing isn't quite so accurate and the colt realizes that the human bonked him, the human might still make the situation a positive one by watching closely. If the colt realizes that the human, like an older horse would do, just nipped or thumped him back, the colt will often move his mouth in a small, repetitive "baby mouthing" movement that essentially says "I'm sorry, don't hurt me." An attentive human might, at that moment reach out with a soft pet down the colt's face in the same smooth manner that his dam's tongue might have stroked him when he was very young, and he'll feel reassured and feel a little more secure in his understanding about how to be around this human.
After letting him lick & chew a moment or two after this, someone practicing "Good Horsemanship" would then lead the colt's attention into the next activity with a positive energy.
Nuthin wrong with that horse that putting the owner in training for a few months wouldn't shape right up.