A power core is a component found in various technologies, responsible for supplying energy with which to power the device or vessel.
In 2369, shortly after Starfleet took over the Cardassian space station Terok Nor, Miles O'Brien, the station's chief of operations, had to effect repairs on the secondary phase modulators of the station's power core, amongst a litany of other technical problems affecting the station. Shortly thereafter, the crew of the station discovered a device connected to the replicator systems which built a virus into the replicated food at the molecular level. Analyzing the device, Major Kira discovered that it contained a power core of diboridium, indicating that it was Cardassian in origin, although it was later discovered that it had been put in place by Bajoran resistance fighters. Later, as the virus swept through the station, Boslic freighter captain Jaheel attempted to escape quarantine by disengaging his ship from the docking port while the mooring clamps where still engaged. In doing so, he inadvertently caused his ship's power core to rupture, threatening a collapse of his fuel cells, which would result in an explosion which would cripple the station. To avert this eventuality, Odo was forced to manually explode the mooring clamps, pushing Jaheel's ship to a safe distance before the explosion. (DS9: "Babel")
In 2376, while engaged in combat with the USS Equinox in the Delta Quadrant, Kathryn Janeway, captain of the USS Voyager, ordered the Equinox's power core targeted with ship's phasers, but they were unable to penetrate the Equinox's shields. (VOY: "Equinox, Part II")
Later that year, the Voyager crew encountered Tash, an alien scientist who had constructed a graviton catapult with which to return to his home planet. The crew assisted with repairs to Tash's power core, in exchange for use of the catapult to shorten their journey to the Alpha Quadrant. (VOY: "The Voyager Conspiracy")