A transwarp conduit is an artificial construction that allows travel over great distances. The opening of a conduit is known as a transwarp aperture.
Between 2373 and 2374, the Borg created a transwarp conduit to the Gamma Quadrant in preparation for an invasion of the Dominion. Their early attempts were stymied by an attack on the assimilated USS Valkyrie by the Jem'Hadar and the emergence of Species 8472 in the Delta Quadrant. (DS9 novel: Lesser Evil)
In 2380, an assembled task force of Federation, Klingon and Romulan vessels attempted to hunt down and destroy the rogue Vulcan T'Uerell who had attempted to form a Borg Collective of her own which used logic to bring about order. They discovered that she had harnessed the power of a polaric ion energy string to help power a highly stable transwarp conduit in order for her to bring reinforcements to protect her. Admiral Kathryn Janeway commented that this conduit was different from the ones she encountered and was unusually stable as it was being directly controlled by T'Uerell using the power of the string to keep it open. This conduit was destroyed once the Vulcan and her starship were destroyed. (ST video game: Legacy)
By the 2390s, the Federation was again experimenting with transwarp conduit technology, based on Borg designs brought back from the Delta Quadrant by the USS Voyager. In 2399, the first links of the Federation Transwarp Network were established, making fast travel between distant star systems possible. (ST website: The Path to 2409)