The transmuter is an amplifier that allows mental impulses to shape the physical world. Explorers Sylvia and Korob used it to enter the Alpha Quadrant. It also enabled them to restructure matter, transport themselves from place to place, exert mind control, and to perform other feats, including some that resembled "magic."
Korob's transmuter appeared in the form of a "magic wand," as he wished to project of himself as a sorcerer. While he never voluntarily released the device he used, it's possible that he could only control it by physical contact. The head of the instrument, a large crystal, glowed when Korob used its power to restructure matter.
Sylvia possessed a less powerful instrument: hers took the form of a crystal pendant, and was the only thing consistent across all her forms. It appeared to be less powerful than Korob's device, as she was not able to use it to alter the fortress-like headquarters he had created. Sylvia also described Korob's device as "holding the key." Whatever the reason, Sylvia was desperate to possess the transmuter after Korob died, using various combinations of trickery and threat to persuade James T. Kirk to surrender it. Ultimately, Kirk destroyed the device, which undid everything Sylvia and Korob had created.
The transmuter also protected the two aliens from some harmful property of our galaxy. Without it, they died in less than a minute.