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 limit 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 Template:ShipClass 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 "upgrades" by Kosinski, the USS Enterprise-D experienced a phenomenal surge of power during a warp speed experiment and was sent over 2,700,000 light years in a matter of minutes. After realizing what he had stumbled upon, by making the so-called warp 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 speed was attained with the help of The Traveler and that Kosinski's formula was bogus. (TNG: "Where No One Has Gone Before")
In 2372, the maximum warp barrier was also known as the transwarp threshold. (VOY: "Threshold")