Transwarp generally refers to speeds and technologies that are beyond conventional warp drives. The warp drive has a natural physical or economical limit beyond which higher speeds are no longer possible. The reference work Star Trek Fact Files indicates this limit at warp factor 9.99. This is the highest conventional warp speed mentioned for a spaceship (Borg cube). Also in the episode Threshold (Star Trek Voyager) the warp factor 9.99 is suggested as the limit. This is the last warp factor mentioned before the leap takes place in the transwarp state.
In the book Star Trek: The Next Generation Technical Manual the authors describes the idea of Transwarp:
Finally, we had to create a back door for various powerful aliens like Q who got the knack of hurling the ship through the room for millions of light years during a commercial break.
The Transwarp concept itself is not tied to any particular technology or speed limit.
The first mention of a transwarp drive took place in the movie Star Trek III: The Search for Spock. There, the Starfleet developed a new spaceship type, the USS Excelsior (NX-2000), which should have a superior engine. The Excelsior captain plans to break the speed record of the USS Enterprise (Warp 14.1 cubic scale). The principle of this drive is not explained. Later, in Star Trek VI: The Undiscovered Country, the USS Excelsior had a normal warp drive. In Star Trek Fact Files it is stated that the experiment was a failure and the spaceship was converted to a normal warp drive.
The entire episode Threshold from Star Trek Voyager is about a transwarp experiment by the USS Voyager crew. To get home faster, a shuttle is modified with novel dilithium crystals. The crew is trying to break the transwarp threshold. This threshold is between warp factor 9.99 and Warp 10 and transwarp itself represented the infinite speed. The shuttle allegedly found itself at all points in the universe at the same time during the flight. However, the pilot suffers genetic mutations after the flight, so it is not repeated. Due to the shuttle’s limited memory, only a small portion of the sensor data was recorded. The entire experiment is described in the reference work Star Trek Fact Files .
Some episodes later, fictionalized a few months later, the crew of USS Voyager encounters a species called Voth. This species has spaceships with transwarp drive. However, this drive does not work on the base of transwarp conduits, as the transwarp drive of the Borg, but is a further development of the conventional warp drive.