(covers information from several alternate timelines)
Containment field generators, or simply containment generators, were critical engineering components of warp-driven starships, responsible for the production of the energy barriers designed to prevent the vessels' antimatter from coming into contact with the rest of the ship. As these field generators were of such critical importance, the loss of even one of them was a potentially catastrophic event.
In 2366 of an alternate timeline, the USS Enterprise-D suffered massive battle damage as a result of combat with three Klingon Birds-of-Prey, including the damaging of containment field generator number three. Although Lieutenant Commander La Forge attempted to bypass the generator, he was unsuccessful, and was forced to attempt an emergency shutdown of the warp core. The timeline in which these events occurred was later negated when the USS Enterprise-C, which had traveled forward in time through a temporal rift, returned to its own time period, thereby correcting the flow of history. (TNG: "Yesterday's Enterprise")
In 2372, the USS Voyager's containment generator was blown out due to an impact by a Kazon torpedo, preventing the ship from escaping at warp until Lieutenant Commander B'Elanna Torres, Voyager's chief engineer, was able to repair it. Voyager suffered the same problem later that year due to another Kazon attack, forcing Torres to attempt a bypass. (VOY: "Basics, Part I", "Relativity")