Supernatural is a term that pertains to any type of phenomena or belief that cannot be proven by modern science or reason, into the realm of the unexplainable events and machinations often observed by superstition, and also some of the faith-based beliefs held by many religions or believers in the arts of magic.
While supernatural phenomena often remain unexplained, as science expands, many events and manifestations formerly associated with magic, superstition and the supernatural could be explained by a greater understanding of natural possibility, thus bringing them into the realm of natural science.
A Human, Arthur C. Clarke, was often quoted as saying that "any sufficiently advanced technology is indistinguishable from magic", with the explanation of the technology being what would determine whether or not the results were supernatural or ordinary in origination. (TOS novelization: Star Trek; ST - Myriad Universes novel: Seeds of Dissent)
Because there was little documentation as to the full extent of the Vulcan civilization's use of various powers of telepathy and telekinesis, these abilities were regarded as mysticism by members of other species, such as Humans. In particular, Commander of Starfleet Harry Morrow was skeptical of issuing orders based on evidence of mind-melds transferring consciousness. (TOS movie, novelization & comic adaptation: The Search for Spock)
Supernatural beings and the afterlife
The afterlife is a common belief in many races and cultures that plays a key part in a number of religions, where any lifeform may expect some form of reincarnation or continued existence after death. Because the afterlife is often described as a disconnect from conventional reality, there are few ways of proving or disproving its existence. Some descriptions of afterlife involve the soul or spirit of a lifeform living on as a supernatural being, or joining a community of supernatural beings. Supernatural beings are often referred to as ghosts or phantoms.
Bajoran religion believed that dead Bajorans survived their bodies as borhyas, often translated as "ghosts". Ro Laren told Geordi La Forge of this when the two were trapped out of phase with the rest of reality, referencing their situation. (TNG episode: "The Next Phase")
Because the Federation made contact with a number of energy beings and non-corporeal lifeforms that could not be detected before the invention of advanced sensor technology, a number of phenomena previously thought to be supernatural beings were later proven to be manifestations of non-corporeal lifeforms.
Likewise, there have been cases of people who have changed into non-corporeal existence through extraordinary means, thus experiencing non-supernatural afterlife. Many naval crews from Earth were transformed into electromagnetic radiation over the course of history by an EM phenomenon that was known to have eventually traveled into space, carrying their disembodied consciousnesses. (TNG novel: Ghost Ship) Similarly, a number of researchers secretly testing transporter technology on Theta Eridani II's moon of Mycena were discorporated, living on neural energy for years, posessing starships and people under their control. Since these manifestations were caused by living beings being converted to energy through explainable and observable means, they were not specifically supernatural in nature. (TOS comic: "Experiment in Vengeance!")
Other types of supernatural beings include those associated with religion and superstition, such as angels, devils, demons, gremlins and banshees
A gremlin was a creature from 20th century Earth folklore. Apparently originating from the experiences of early airplane pilots and engineers, a gremlin was a mischievous being that created mechanical problems and failures that could not otherwise be explained. The existence of gremlins was still espoused—usually in a tongue-in-cheek manner—into the age of spaceflight and through the 24th century. (ENT episode: "Dead Stop", VOY episode: "Live Fast and Prosper", CoE eBook: Troubleshooting)
In 2275, Montgomery Scott described the problems with the Sternn Fusion Core on Nova Empyrea as phantoms, as he could not determine the causes for the mechanical failures. One superstition that Scott had developed a belief in was that of gremlins. By all indications, Scott succeeded in trapping such a being in 2286 after the USS Enterprise-A was plagued by a series of equipment failures. Scott placed a well-tuned internal combustion engine in an empty torpedo casing, and when the engine began to fail, Scott launched the torpedo towards an out-of-service space buoy in order to keep the being occupied while the ship moved away. Because of the mechanical failures, no records of the nature of the being were made. (TOS novel: The Better Man; TOS comic: "The Corbomite Effect!")
Magic and supernatural powers
The term "magic" is usually used by modern Humans to describe sleight-of-hand and trickery to create a belief that supernatural events could occur, in a magic act. Such practices of magic are considered quite entertaining when they amaze audiences, even if the magic is not completely convincing. The prospect of "real" magic is not commonly believed, with many earlier suppositions of supernatural magic being explained scientifically in good order. (TOS novel: Devil World; TOS comic: "So Near the Touch")
Magic is a key force in a reality where Megas-Tu is located. All beings there practice varied magic on a daily basis, including acts of telekinesis, teleport, transmutation, and enchantment. When denizens of "normal" reality visit Megas-Tu, they inherit these powers as a natural function of that universe. (TAS episode & novelization: The Magicks of Megas-Tu)
Extrasensory perception is often thought of as a supernatural ability, but Federation science has shown that there is scientific proof that the abilities are natural, since some species have much greater, and more measurable command of these powers. A correlation was found between ESP abilities and the psionic energy of the galactic barrier when several espers found themselves attracting emissions of that energy. (TOS episode & novelization: Where No Man Has Gone Before)
Vulcan telekinesis and transmutation abilities were thought to be an aspect of their mythology for many years, after old mind-skills were forgotten. Some of these abilities made a resurgence when the katra of Zakal was stolen from isolation and allowed to inhabit a body. Some of the principles of Vulcan telekinesis were demonstrated for Federation observers when a terrorist group assembled the Stone of Gol, a powerful psionic amplifier that focused a Vulcan's natural psionic energy to create a powerful telekinetic effect. (TOS - Rihannsu novel: The Romulan Way; TOS - The Lost Years novel: The Lost Years; TNG episode: "Gambit")