Transwarp is an extremely fast form of propulsion, utlizing transwarp conduits that have different properties than standard space. It is the main propulsion system for the Borg, although Federation scientists have studied the principles involved with creating a transwarp drive, with marginal success.
Fundamentally, transwarp drive is a generic term for technologies that overcome the limitations of conventional warp drive.
In 2369, the Starfleet X-Project saw to the development of another experimental starship with transwarp ability. The ship, the USS Phoenix-X, would be able to create its own temporary conduits for faster than warp speeds. Unfortunately, the X-Project was reported to be another failure in 2374; when in actuality, this was merely a cover-up by Section 31. Section 31 hoped to utilize the vessel for their own missions. The Phoenix-X was then launched in service of Starfleet in a supposed normal warp drive. (Star Trek: Phoenix-X)