To me, natural consequences means that you, the parent, step out of the way, and allow the situation to unfold on its own, so that the child experiences what happens automatically as a result of his/her actions. In other words, I, the parent, don't punish, but I don't rescue the child from the results of his actions.
For example-- DS throws his bowl of cereal on the floor. It stays there, and he gets no more cereal, because there's no bowl to put it in.
DD2 drops her pony down the stairs. The pony breaks, and it stays broken.
DD1 dawdles when she's supposed to be getting ready for bed. Bedtime arrives, and there's no time left for a story.
(All of these have happened to us in the last day or so..)
I think that natural consequences are the most effective learning tools for children, as long as the consequences that result from their actions are within their capacity to handle. Like the natural consequence of running out in the road is getting hit by a bus, but clearly the parent cannot just stand back and allow that to happen. If a child refuses to have his diaper changed, the natural consequence of course would be to get a horrible diaper rash, but that's not a consequence most of us would be willing to allow. But when a consequence is not life-threatening, not unhealthy, and not irreparably damaging, you stand back and let the child experience the real-world results, and learn from them.
This can be hard, because we have an urge to protect our children from disappointment, from unhappiness, from hard work, from frustration, etc. It's so much easier to just pick up the bowl, or read the story anyway, or fix the pony, than to watch the child go through the anger and disappointment. But these kinds of "consequences" are such powerful learning opportunities.