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 2254, when the crew of the USS Enterprise discovered what they thought were survivors from the SS Columbia, which had crashed on Talos IV in 2236, José Tyler noted that the survivors would not believe how fast they could get back to Earth, due to the breaking of the "time barrier" that had occurred since the Columbia had crashed. (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")