When you pray, the words and thoughts are believed to achieve potential results. This is no different from a wizard or a pagan casting a spell and some books routinely include prayer when discussing magic1. It is supernatural because the effect is not achieved through the physical laws of nature that can be investigated through science. Magic is simply a form of supernaturalism bought about via ritual or spoken words. Magic is not taken seriously by scientists, academics, skeptics nor the general populace. Despite amazing and great advances in physics, quantum physics, psychology and neurology, no possible basis for "magical" actions has been found by researchers. All sociological investigations have found no evidence of real magical power, and parapsychological and occult experimenters have never been able to formulate proof that satisfies basic scientific requirements such as independent verification and impartiality. So, in the modern world, all major religions disclaim themselves against magic and say that they don't practice it. Ask a Christian if praying to heal someone is magic, or a Muslim if the healing ruqyah is magic, and they'll vehemently deny it. 2
Many others would call it magic. The actual method by which a ritual works is unknown. It is the same with prayer: When people think that prayer has been successful, they ascribe the cause and effect of it to God. But this cause and effect could be any supernatural phenomenon, such as telekinesis or psi. It could be that the spirits of the dead carry out the wishes of a Christian who prays (therefore, folk religion is true, and Christianity is false). It could be that everyone, including Christians have spiritual animal guides, and that these oversee people's wishes and prayers, and go forth and make some such wishes actually happen. We don't know if spells, prayers, rituals, and wishes are made effective by quantum souls, gods, fairies, or demons. If they work via a God, then, we simply don't know which God is doing the granting. We simply do not know what happens; hence, prayer is not only supernatural but it is magic. When religious people hear how other people explain these effects, they turn into hardcore skeptics. It is magic, and, magic is daft. But when they are talking about their own magical pursuit, known as prayer? Well, in this case, they think, it is not magic, and it is not daft! But how does it work? They still don't know. There is no different between the magical words of prayer, and any other form of magical word. Prayer is magic.
The Jehovah's Witnesses (JW) managed to hit the nail on the head when it comes to their description of magic - a description which fits prayer perfectly!
“In its most basic sense, magic is an effort to control or coerce the natural or supernatural forces to do man's bidding. Not knowing the real cause of many everyday happenings, people in earlier societies believed that the repetition of certain magical words or incantations, or the performance of some ritual, could bring about certain desired effects.”
"Mankind's Search for God" by The Jehovah's Witnesses (1990)3
The entertaining part is that the JW text was writing against pagan practices, and not against the prayers of Christians. They seem unable to apply the same logic they use to criticize others' practices against their own. This kind of double-think underpins the modern theistic insistence that prayer is not magic.
People don't really believe that God can, or will, do anything as the result of prayer. It seems odd, given that many people think they believe in prayer, to announce that many of them don't! A series of studies by Justin Barrett focused on asking people which of three things they might pray for as a result of an imminent disaster5. For example, if a ship in the middle of an Ocean was struck and was sinking, would they pray for (a) God to enable the ship to float for longer with a broken hull, (b) for God to let the people survive for longer in the deadly icy cold waters, or (c) for a nearby ship captain to have a sudden idea to change course and therefore rescue the people. Most people pick the third type of prayer: this is a prayer in which human action saves the day - God doesn't interfere with the physics of the ship nor with the biology of the victims of the accident5. A change in human action saves the day. This is because those who pray subconsciously think that it is their own magical words that are enacting the change, and not the creator of the universe.
“Barrett was careful to make all these possible interventions equally salient, and though they are all trivial for an omnipotent god, most subjects spontaneously chose the third kind of option [... as] it is easier for a person [who is praying] to change people's mind than to correct or reorient physical and biological processes. But note that this expectation would be irrelevant if God's great powers were the most salient [facilitator].”
See: "The God of the Gaps" by Vexen Crabtree (2010).
Religionists are apt to describe confusing unknowns as the work of God. Physics and biology have long since been thoroughly understood by science6 and have lost any mystical or spiritual appeal. No longer do gods push the sun and moon around the sky, nor cause thunder. The human decision-making process though, because neurological studies in cognitive psychology are not well known (or understood) by the populace, means that our culture still retains the idea that God can play a part in this process. This is the "god of the gaps" phenomenon. Deep down, many theists understand that God's power is not real; but they still retain the secret, subconscious and superstitious belief that the magical words and the well-wishing of a prayer might still be powerful enough to change a human's actions, even though they don't think prayer could change purely physical processes. When it comes to complicated biological factors where there are statistics involved (i.e., cancer and disease), many theists again think that prayer might work. This is especially true for minor, non-biological ailments, where psychological effects play a big role (i.e. psychosomatic effects), meaning that human agency is still felt to be important (and manipulatable).