Supernatural (and/or paranormal) typically refers to claims of phenomena which are not bound to natural laws or observability. Because of this, science cannot and does not attempt to explain these phenomena, as they are untestable and cannot substantiate evidence.
Throughout human history, there have been many claims of supernatural events or supernatural abilities. No claims have ever been demonstrated to to be true. Often times, claims are hoax's, and are debunked (James Randi exposing Uri Geller, for example).
In other cases, claims are simply not falsifiable, and testing them has no purpose. For example, prayer. If one prays for X, and X happens, they can attribute this to whatever figure they prayed to. However, if X does not happen, it is often rationalized by stating "it wasn't ready to happen yet" or "This means I'm focused incorrectly, and don't need it". With these rationalizations, prayer is not falsifiable, and becomes indistinguishable from coincidence or chance. Often times, one can replace the figure they prayed to with any other thing (for example, a brick or carton of milk), and the "argument" for prayer now "demonstrates" a completely different thing.
Often, people who believe in one form of paranormal activity will not believe in others. Thus a devoutly religious person may regard ESP or witchcraft as nonsense. This is a very useful fact to keep in mind when debating the issue: if they can understand why they don't believe those claims, they can better query their own beliefs.