Transwarp basically creates a
Wormhole.
My made-up-on-the-spot description is that you make a hole in space with plain
warp drive and fall into it but with Transwarp you're creating a
tunnel and you fall
through it.
Used by
The Borg.
Child's Play (VOY):
Star Trek: Voyager, the scourge of all Star Trek, has apparently decided that transwarp conduits are permanent, kind of like Subspace Conduits ("under-space") from Dragon's Teeth (VOY). I guess you need a transwarp drive to open them so you can traverse them. *rolls his eyes*
As Gordian cleverly points out, in Star Trek III the U.S.S Excelsior had an experimental transwarp drive. It's generally accepted, AFAIK, however, that it was a failed experiment. The Star Trek Encyclopedia, which is, I believe, officially considered canon (that's an odd phrase) too says that the experiment was a failure.