The galactic core, or galactic center, is the central region of the Milky Way Galaxy. Anchored by a supermassive black hole, it is subject to intense radiation and increased risk of astronomical near-misses (resulting in climate change on the local planets) which makes life unlikely to prosper in the region. The core is surrounded by an immense energy field called The Great Barrier. The core had a diameter of 10,000 light years.
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 miscalculation 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)