Development of the backstory
Warp drive has been a feature of Star Trek since it started. The first pilot episode of Star Trek: The Original Series, "The Cage", calls it "time warp" drive, and notes that the "time barrier" had been broken, allowing a group of stranded interstellar travellers to get back to Earth much quicker than they had been previously able to.
The episode "Metamorphosis", from the original series, establishes a backstory for the invention of warp drive, stating that it was invented by Zefram Cochrane of Alpha Centauri. Cochrane is repeatedly referred to afterwards, but the exact details of the first warp trials were not shown until the second Star Trek: The Next Generation movie, Star Trek: First Contact. The movie depicts Cochrane as inventing warp drive on Earth in 2063 (two years after the date speculated by the first edition of the Star Trek Chronology). He used the immense power given off in a matter-antimatter reaction to give energy, which he could use to move a ship into a subspace warp bubble that could then move the ship at faster than the speed of light. This directly led to the first contact with the Vulcans.
The later prequel series Star Trek: Enterprise firmly establishes that many other civilizations had warp drive before humans, notably the Vulcans, who had more advanced warp drive technology than humans even in the 22nd century. Enterprise, set in 2151 onwards, shows the voyages of the first Earth ship to be capable at going at warp factor 5 which under the old warp table formula was taken as one hundred twenty-five times the speed of light. Using that formula the velocity was the warp factor cubed times the speed of light or 5³*186,000/sec or 5³c, which is 23,250,000 miles per second. This velocity would allow a Federation Starship traveling from Earth at a constant warp 5 to reach Proxima Centauri, Earth's closest stellar body at 4.25 light years (ly) (4.25 X 6 trillion miles) distance (Alpha Centauri is 4.36ly away) in approximately 12.41 days. By the time of Captain Kirk's era in mid 23rd century, Warp factor 8 was within the capabilities of Starships (which was not exceded often due to the strain placed on the engines). Warp 8 was 8³c (512c) or 95,232,000 miles/sec. This would allow travel from Earth to Proxima Centauri in only 3.029 days, approximately the same amount of time it took Apollo 11 to travel from the Earth to its Moon in July 1969.
The Next Generation era
Plots involving the Enterprise going far too fast were a frequent feature in the original series, and for The Next Generation, it was decided that these would no longer be featured. A new warp scale was drawn up, with warp 10 set as an unattainable maximum. This is described in some technical manuals as Eugene's Limit as an homage to creator/producer Gene Roddenberry.
The warp factors above warp 10 in the TOS, such as the one above, were slower than warp 10 on the new scale. According to The Star Trek Encyclopedia, warp 6 (new scale) is equal to 392c (392 times the speed of light, c) and about warp 7.3 on the old scale, whereas warp 9.2 new, to about 1649c and warp 11.8 on the old scale. Under this new definition warp 9.2 translates to 306,714,000 miles/sec. Travel to Proxima Centuri from Earth would only take 22.58 hours.
The scale reaches an asymptote at warp 10 which represents infinite speed in accordance with the speed limit imposed by the producers. The Star Trek: Voyager episode "Threshold" agreed with this, in that the characters said attaining the velocity of warp 10 was impossible (called Eugene's Limit, another homage to Roddenberry) — but then they achieved it anyway, with the side effect that they hyper-evolved (reversibly) into anthropomorphicnewts. In this episode, Tom Paris describes that, while travelling at warp 10, he is concurrently in every part of the universe. At this speed, the Shuttlecraft Cochrane's sensors are able to process enormous amounts of telemetry such that the data storage of the shuttle is completely filled.
The limit of 10 did not entirely stop warp inflation. By the mid-24th century, the Enterprise-D could travel at warp 9.8 at extreme risk, while normal maximum operating speed was warp 9.6 and maximum rated cruise was warp 9.2. The Intrepid-class starship Voyager could manage warp 9.975. The old and new formulas are explained in much greater detail below in the section "Warp Velocities".
The alternate future depicted in the Next Generation episode "All Good Things..." shows Federation vessels capable of going warp 13 when Admiral Riker, commanding the Future Enterprise-D, uses this extra turn of speed to rescue the crew of the USS Pasteur. However, this episode was produced before the Enterprise-D was destroyed in Star Trek: Generations, so the two universes may diverge further than previously expected, and warp 13 may not be possible in the "real" Star Trek universe. It is unclear whether the warp 13 achieved in the possible future shown in "All Good Things..." represents a new recalibration of the warp curve or some form of transwarp, though as this future was a creation of Q it might not occur in the "real" Star Trek timeline.
Transwarp
The term transwarp has been used a number of times, referring to an advanced form of warp drive most commonly used by the Borg, but also the subject of a Starfleet development project in Star Trek III: The Search for Spock.
Episodes of TNG and Voyager seem to indicate that transwarp is best described as a wormhole-style conduit through subspace: this suggests a subsuming into subspace, rather than warping normal space via subspace.
However, in the Voyager episode "Distant Origin", a species known as the Voth used a transwarp technology didn't appear to be similar to Borg transwarp, but rather an enhanced warp technology.
Borg Conduits
The Borg (in the The Next Generation two-part episode "Descent" and in the Voyager finale Endgame) have discovered the existence of transwarp conduits—regions in subspace that facilitate transwarp travel at up to 20 times faster than conventional warp drives. These episodes established that the Borg set up networks of these conduits between important areas in the galaxy. Borg transwarp conduits are activated by an encoded tachyon pulse. When a Borg vessel enters a transwarp conduit, it is subject to extreme gravimetric shear. To compensate, the Borg project a structural integrity field ahead of the vessel. Artificial conduits are linked together with transwarp hubs. Six hubs were known to exist, but in "'Endgame" one was destroyed, along with the Unicomplex due to the neurolyticpathogen with which Admiral Janeway infected herself.
Quantum Slipstream
See Slipstream (science fiction)
Quantum Slipstream Technology is presumed to be the standard interstellar propulsion method used by Species 116 (of which Arturis was a member) prior to their assimilation by the Borg. In the Voyager episode "Hope and Fear", Seven of Nine remarks that the technology involved is not dissimilar to Borg transwarp technology.
Warp velocities
Warp travel velocity in Star Trek is generally described in "warp factor" units, which - according to the Star Trek Technical Manuals - correspond to the strength of the warp field. Achieving warp factor 1 is equivalent to breaking the light-speed barrier, while the actual speed of higher factors is determined according to an ambiguous "warp formula". Several episodes of the original series placed the Enterprise in peril by having it travel at high warp factors; in "That Which Survives", this factor was as high as 14.1. However, the actual speed of any given warp factor is rarely explicitly stated on screen, and travel times for specific interstellar distances are not consistent through the various series.
According to the Star Trek episode writer's guide for The Original Series, warp factors are supposedly converted to multiples of light speed with the cubic functions(w) = w3c. Accordingly, "warp 1" is equivalent to the speed of light, "warp 2" is eight times the speed of light, "warp 3" is 27 times the speed of light, and so on. However, this conflicts with the on-screen application of the technology, as it would make the Enterprise far too slow for the voyages depicted in the television series. These speeds do not even correlate with details presented in some of the episodes. For example, in "That Which Survives" (1969), the Enterprise travels at warp 8.4 for 11.33 hours and traverses 990.7 light years (as indicated in Spock's dialog), which makes the speed more than 600,000 times the speed of light. The Enterprise has also easily traveled to and from the edge of the Milky Way galaxy ("Is There in Truth No Beauty" and "By Any Other Name" (1968)), a journey which should take years at "warp 8" if the actual speed is merely a cube of the warp factor.