The Galactic barrier is a mysterious energy field of unknown origin, surrounding the Milky Way Galaxy (or at least the near side of it). It manifests as a pinkish-purple band of energy, which has been encountered by numerous Earth and Starfleet vessels. One of the closest planets to the barrier is Delta Vega.
Two centuries later, in 2265, the USS Enterprise discovered the Valiant's recorder-marker and learned that the Valiant's captain had destroyed the ship himself, to prevent a mutated crewmember from escaping.
The Enterprise breached the barrier herself, resulting in crewmen Gary Mitchell and Elizabeth Dehner gaining "god-like" powers. Mitchell would have destroyed the ship and threatened the entire galaxy had he not been killed by his best friend, Captain James T. Kirk, with Dehner's help. (TOS: "Where No Man Has Gone Before")
In 2268, the Enterprise twice encountered the barrier; once when hijacked by the Kelvans, and again when engineer Lawrence Marvick was driven mad by the sight of Medusan Ambassador Kollos. (TOS: "By Any Other Name", "Is There In Truth No Beauty?")
Over the next century, other Starfleet vessels would also encounter the barrier, including the USS Deimos in 2301, the USS Stargazer in 2333, and the USS Enterprise-E in 2374. (Starship Deimos: Where They Have Gone, We Follow; TNG novels: The Valiant, The Q Continuum)
While on a survey of the galactic rim in 2383, the USS Pendragon was diverted to Delta Vega to recover the bodies of Mitchell and Dehner. Exposure to the residual energy in the bodies altered Lt. Commander Ben Bartholomew, and ultimately resulted in the Pendragon breaching the galactic barrier. (Star Trek: Pendragon: "Whispers As Loud As Thunder")
The barrier was once again encountered in 2384, this time by the Phoenix-X, on a dual mission to complete the X-Project testing and use it to travel to the Andromeda galaxy. Upon a rough crossing of the barrier, through the use of ultra-transwarp, the crew were momentarily met with the being 0. Luckily, Q had been traveling aboard the Phoenix-X and was able to deal with the other omnipotent being. (Star Trek: Phoenix-X: "Forbidden Crossing, Parts II & III")