Vulcans using telekinesis
The fal-tor-pan
Mysticism referred to a variety of different religious beliefs and practices, often associated with magic or some otherworldly supernatural forces.
In the case of the Vulcan people, it referred to certain practices such as their unique mental abilities, such as psychokinesis, telepathy, and their ability to transfer their own katra to other host bodies.
The women of Elas were described by Federation scientists as having mystical powers, making them able to "drive men wild." (TOS: "Elaan of Troyius")
In 2285, Admiral James T. Kirk explained to Admiral Morrow that Spock's katra was locked inside Doctor Leonard McCoy's body, and the only way to free it was to return him to Vulcan. Morrow refused to hand over the USS Enterprise to Admiral Kirk, and in addition to the fact that he felt the Enterprise would never take the pounding involved in making the trip to Vulcan, he told him that he didn't know if he believed in Vulcan mysticism. (Star Trek III: The Search for Spock)