(covers information from several alternate timelines)
This page contains information regarding Star Trek: Discovery, and thus may contain spoilers.
The galactic core or galactic center was the central region of the Milky Way Galaxy. The core was surrounded by an immense energy field called the Great Barrier. (Star Trek V: The Final Frontier) The galactic center was relevant to starship navigation, as Starfleet ships could verify their position relative to it. The heart of the galaxy itself was at galactic coordinates 036.231. (DIS: "Despite Yourself"; TAS: "Beyond the Farthest Star")
In 2269, the USS Enterprise, while on a scientific mission to find the creation point in the galactic core, found an entrance to an alternate universe where magic replaced science. (TAS: "The Magicks of Megas-Tu")
During the Federation-Cardassian War, Cardassian Central Command claimed a Federation stellar cartography map was in error, due to a miscalibration by a reference beacon. They maintained that the Federation-Cardassian border in several sectors should be adjusted 4.7 light years to the galactic center. (TNG-R: "Chain of Command, Part II")
In 2367, the USS Enterprise-D was temporarily transferred near the center of the galaxy by Lieutenant Barclay, where an intelligent species known as the Cytherians was discovered. (TNG: "The Nth Degree")
Later that year, the galactic core was seen on the map "USS Voyager Flight Path". According to this map, the portion of the core located in the Alpha Quadrant had been explored by manned and unmanned probes. (Star Trek: Voyager, Season 7 set artwork )
After a spore drive jump, the USS Discovery would confirm their coordinates relative to the galactic core. After jumping to the Sol system in 3189, Commander Michael Burnham confirmed coordinates relative to the galactic core to Captain Saru. (DIS: "People of Earth")