A folded space drive was an experimental type of advanced interstellar propulsion system. Federation scientists Drs. Henrik and Samantha Delaney were working on such as system for Starfleet, and, by 2371, they created a stable prototype for testing.
The folded space drive worked by "folding" time-space, making the journey between two points quicker by bringing those points in space closer.
The Delaneys and the folded space drive prototype
In 2371, Starfleet installed the first prototype aboard the USS Rutherford, an experimental type of starship. Initial tests, monitored by Henrik Delaney, were successful; however, late in the tests, the drive malfunctioned, sending the Rutherford to the Gamma Quadrant and stranding the crew there.
In 2372, investigations by the USS Prospect-A crew found the folded space drive prototype missing from the unmanned Rutherford. In addition, sensor readings on New Bajor indicated Henrik Delaney triggered the device to allow him and other survivors to escape (and to prevent the prototype from falling into the possession of) the Jem'Hadar. (Star Trek: The Prospect Chronicles: "Directive")
In that same year, Hahn Jun-Seok, an undercover Na'arbi agent, visited Samantha Delaney on Starbase Expanse 12, a top-secret testing facility in the Devil's Cradle near the Typhon Expanse. Once there, he wiped all data about the folded space drive from the computers.
Later in 2372, driven insane by the loss of her husband, her children aboard USS Voyager and the loss of her years of hard work, Samantha Delaney created another two prototypes: one to use as a weapon and another to use to escape with. (Star Trek: The Cantabrian Expeditions: "An Innocent Time")
A folded space drive, when misused, could tear holes in the fabric of time-space, resulting in the overlapping of time and space as well as alternate dimensions. The second-hand effects of misuse of the drive included insanity. The folded space drive could also be used theoretically to shift a starship from one dimension to another. (Star Trek: The Cantabrian Expeditions: "An Innocent Time", "Salvation")