The computer core housed primary processing equipment including the computer core processors. 23rd century Starfleet ships had duotronic systems, while 24th century ships had isolinear cores. The computer cores communicated with each other and ship's systems via an optical data network (ODN).
The Galaxy- and Excelsior-classes of Federation starships carried three computer cores, while ships of the Intrepid- and Sovereign-classes contained two. These multiple cores provided redundancy in case one of the cores was damaged or destroyed.
In 2367, Lieutenant Reginald Barclay created a neural scan interface that communicated his thoughts directly to the USS Enterprise-D's central processing unit. This meant much of his higher brain functions and memory were transferred to the starboard computer core while his body remained in the holodeck. His primary cerebral functions were operating almost entirely from within the computer, and had expanded to such a degree that he determined it would be impossible to return to the confines of his human brain without resulting in his death. (TNG: "The Nth Degree")
In 1996, Henry Starling managed to download almost twenty percent of the USS Voyager's computer core to his personal computer by using the ship's transporter beam as a downlink. (VOY: "Future's End", "Future's End, Part II")
In 2293, the location of the computer cores aboard the Excelsior-class USS Enterprise-B were labeled in a MSD. Two of these cores were located in the primary hull and a third was located in the secondary hull. (Star Trek Generations)
After beaming aboard the IKS Drovana to obtain the detonation codes for a Klingon minefield in 2372, Worf and Kurn told an officer they were running a diagnostic on the starship's computer core. (DS9: "Sons of Mogh")
That same year, Commander Chakotay noted that a piece of organic technology aboard a Species 8472 bio-ship looked like a binary matrix but it was laced with neuropeptides, which he further speculated to be its version of a computer core. (VOY: "Scorpion")
In order to take Deep Space 9's computer systems offline and prevent the Dominion from destroying a minefield covering the Bajoran wormhole, Kira Nerys and Rom considered planting a bomb in the station's computer core. They later attempted to sabotage it by more standard means. (DS9: "Sacrifice of Angels")