Transwarp is a term for any speed that exceeds that of traditional warp drive limitations.
Federation Transwarp Experimentation
Efforts to develop a transwarp propulsion system date back to at least the 2260s. A research conducted over eight years into the field of transwarp was made by a specialist in interspace physics called Perren. In the mid 23rd century, Professor Ursula Mornay of the University of Tarrigor perfected the theory for transwarp though Perren was responsible for constructing the actual hardware needed to make such an engine. In 2270, one of the first experimental transwarp engines was constructed. However, it nearly fell into terrorist hands. (TOS novel: Battlestations!).
In 2284, a transwarp engine jointly developed by Starfleet and the Daystrom Institute was tested by Trill pilot Torias Dax. Though Torias died during a test flight, project head Admiral Harrison Morrow allowed the project to continue. (DS9 short story: "Infinity")
By 2285, the USS Excelsior served as the testbed vessel for the Transwarp Development Project. Unfortunately, the experiment was ultimately deemed unsuccessful. The Excelsior was refitted with standard engines and given the registry NCC-2000. (TOS movie, novelization & comic adaptations: The Search for Spock, The Undiscovered Country)
The failed transwarp technology was employed again in 2311 as part of a ruse against the Romulan Star Empire. The USS Universe was presented as a testbed for an experimental "hyperwarp drive." In fact, the ship was fitted with transwarp engines, and its energies directed through a Romulan cloaking device in order to create a unique energy signature. The Romulans mistook these readings (and the explosion of the Universe) as evidence the Federation was developing a metaweapon. These events precipitated the Tomed Incident and the ratification of the Treaty of Algeron. (TLE novel: Serpents Among the Ruins)
Despite the failure of transwarp, Federation advances in warp theory lead to a redefinition of the warp scale, establishing warp 10 as infinite velocity. (By the 24th century, maximum warp speed was warp factor 9.975 (3053 times the speed of light), which could only be sustained for a few hours by the fastest Federation starships.) In 2372, Lieutenant Tom Paris succeeded in exceeding warp 10 into transwarp, but the subsequent mutagenic effect the process had on his DNA prevented anyone from following up that line of exploration for the foreseeable future. (VOY episode: "Threshold")
In an alternate timeline created by the War of the Prophets which started with the formation of the Red Wormhole, the Federation had developed advanced transwarp technology by the year 2400. This was accomplished primarily through the alliance made with the Borg in the Treaty of Wolf 359. Ships possessed variable transwarp speeds which were identified as "transfactor" speeds. (DS9 - Millennium novel: The War of the Prophets)
The Borg employ a series of transwarp conduits, joined by transwarp hubs, by which they are able to cross vast distances across the galaxy. (TNG episode: "Descent"; VOY episode: "Endgame") Variant uses of the technology were seen in both the Transwarp Gate and Transwarp Portal. (ST video games: Armada, Armada II)
In the Star Trek Online time frame transwarp technology is fitted on all Federation and Klingon Empire starships. The technology is limited by only being able to use transwarp to go to Earth Spacedock for the Federation or Qo'noS for the Klingon Empire. A prototype of a transwarp drive, as seen on the Excelsior, later allows travel to various Sectors of Space. Members of the Federation Diplomatic Corps have access to this technology to go to Deep Space Station K-7, Starbase 39-Sierra and Deep Space 9.