Most people's working definition of the supernatural is something like: anything really neat that I cannot understand and which seems impossible. Harry Potter raises his magic wand, shouts "Expelliarmus," and a flash of lightning sends his opponent's wand flying across the room. That certainly meets the working definition of supernatural. I have no idea how that could have happened, and it sure looks impossible.
But, a 747 taking off arguably meets that definition, too. I don't know airplanes work, and all that metal gliding through the air sure looks impossible to me. Yet no one calls airplanes supernatural. A couple of centuries ago, people might have agreed that a 747 taking off was a supernatural event, had they chanced to see one departing from the local airport, but not today. Now it is commonplace, although a 747 leaving London Heathrow on time does still seem a bit miraculous to anyone who is familiar with that particular airport.
Our working definition of supernatural leaves something to be desired because it is based on what a particular person knows about how the world works. If you don't know how airplanes fly, they are supernatural to you, but if you do know, or perhaps even if you just know that someone else knows, then airplanes are no longer supernatural. The working definition is as much about the observer's state of mind as it is about the allegedly supernatural phenomenon. By that definition, the supernatural clearly exists. It just changes all the time.
So let's get a bit more serious and look at the actual definition, which is something like: outside of nature, or not following the laws of nature. By that definition, or any reasonable definition, the concept of the supernatural is barely coherent. It is easy for someone to say they believe in the supernatural, but if looks at the concept a bit closer, it simply loses meaning. The supernatural does not exist, by its very definition.
For example, let's say that we actually see Harry Potter disarm his opponent by pointing his magic wand and shouting the magic word, "Expelliarmus." Such an event is indeed supernatural -- until it actually happens. Once it happens, it is, by definition, natural. It might be new and surprising, but that does not matter. At one time, flight was new and surprising. For centuries, people believed that the laws of nature simply did not allow bigs hunks of metal to fly through the air. But they were wrong. That does not mean that the laws of nature changed. It simply means we did not fully understand them before. And of course, we do not fully understand them now, either.
The laws of nature are not like the laws of the United States or the laws of Britain. The laws of the United States are only laws because Congress passed them and the President signed them. Those laws can change. Moreover, they can be broken. That is what prisons are for. And, it is even possible to break the law and not go to prison. The laws of nature work in a completely different fashion. Scientists do not write down laws and command nature to obey. Rather, they observe what happens, and try to come up with rules that explain what is happening. Those are not "laws" in the same sense as the laws of the United States.
The laws of nature cannot, by definition, be broken. If the laws of nature as calculated appear to be broken, then it is laws that are wrong, not nature. And of course that happens all the time. The most powerful laws of nature, like F=MA, often turn out to be approximations that only "apply" under certain conditions. Einstein's theory of relativity is a better approximation, and made predictions that turned out to be correct, such as the bending of light in a gravitational field. That bending of light might appear "supernatural" without knowledge of relativity.
When people imagine supernatural events, they imagine them to follow logical rules. For example, Harry's spell would not have worked if he were not a wizard. It would not have worked if he had used a cricket bat instead of a magic wand. It would not have worked if he had said the magic word incorrectly. Ghosts do not randomly appear in the middle of the day, in the middle of the woods, with no one around. Rather, they haunt the houses where they were horribly murdered, in the middle of the night, because it is angry. The fortune teller follows fixed rules to interpret the cards. Perhaps those rules are new, strange or not fully understood. But they are still rules. (And, if those rules were chaotic, that still wouldn't matter. There is nothing that says that nature must be orderly, although it has always proven to be. I'm not going to follow that line of thought here).
Some may be tempted to try to rescue the supernatural by changing the definition to something like: things that humans can never fully understand. But that definition is not terribly interesting, and it certainly does not capture what people really mean when they speak of the supernatural. Those who know the most about quantum mechanics say that they do not really understand it, and never will. Richard Feynman, for example, admitted that he did not really understand quantum mechanics. Yet no one claims that quantum mechanics is supernatural. It is just really hard to understand.
Of course, Feynman could describe quantum mechanics, just as J.K. Rowling can describe Harry Potter and his magic tricks. It is easy to say: This is what I saw happen. The electrons are behaving as if they are both here and not here at the same time. I saw blood flowing from the eyes of the statue. I saw him lifted up to heaven. She was able to read my thoughts. These are all supernatural events by any meaningful definition, and they can all be described in the sense that we can report what happened. The fact that we can report on an event does not mean we understand it, and if we cannot even understand something well enough to report on it, we cannot even begin to have a discussion about it.
Moreover, there may be things even more complicated than quantum mechanics. Some humans, like Richard Feynman, can understand quantum mechanics in the sense that they can show it is true and describe what occurs in mathematical terms. Perhaps there are natural phenomenon even more complicated, such that our minds are simply incapable of comprehending. But that does not make those events supernatural. It just means our minds are limited, something we knew already.
Anything that actually happens is part of nature, whether or not we understand how it works. If something does not happen, then it is not part of nature. The supernatural cannot, by its very definition, exist. So, in the end, belief in the supernatural comes down to belief in gibberish. No one can really believe in the supernatural if they think about it in any meaningful way.
So, why do people insist on saying that they believe in the supernatural? Because people want to believe. Because it is exciting, scary -- or, in the case of religion, reassuring -- to believe in something being "out there" that is beyond our understanding. Harry Potter would not be magical if we knew the physics behind his tricks. Thinking about it to hard -- or reading pedantic blog entries like this one -- takes all the fun out of the supernatural.
In the end, we are just going back to that old working definition of supernatural: anything that we do not understand and which seems, in our experience, to be impossible, but with a slight twist: the supernatural is something we want to believe.