The Phoenix breaking the warp barrier
A schematic of the Malcorian warp ship
The warp barrier was the transition from sublight velocities to faster-than-light warp speeds. It was also used as a reference to the limits of warp drive technology to achieve certain rates of speed.
In 2063, the Phoenix reached warp threshold and broke the warp barrier. (Star Trek: First Contact)
In 2143, the NX-class prototypes broke the warp barrier, known as the warp 2 barrier. (ENT: "First Flight")
The time barrier was a warp barrier broken some time between 2236 and 2254. (TOS: "The Cage")
In 2364, after a series of formula modifications by Kosinski, the USS Enterprise-D experienced a phenomenal surge of power during a warp speed experiment and jumped over millions and later over a billion light years through space in a matter of minutes. During the first jump, instrumentation indicated that the ship actually surpassed warp 10, apparently making the so-called warp 10 barrier meaningless. Kosinski declared that science would need new definitions and new parameters to define warp speed. One suggestion was the Kosinski scale. It was later determined that the enormous surge of speed was caused by The Traveler and that Kosinski's formulas were bogus. (TNG: "Where No One Has Gone Before") The Traveler had the ability to alter warp fields using the energy of his thoughts to move through space and time. (TNG: "Time Squared", "Remember Me")
In 2372, the warp 10 maximum warp barrier was also known as the transwarp threshold. (VOY: "Threshold")