The computer core houses primary processing equipment. 23rd century Starfleet ships had duotronic systems, while 24th century ships had isolinear cores. The computer cores communicate with each other and ship's systems via optical data network (ODN).
The Galaxy-class of Federation starships carry three computer cores, while ships of the Intrepid- and Sovereign-classes contain two. These multiple cores provide redundancy in case one of the cores should be damaged or destroyed.
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")
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")
In order to take Deep Space 9's computer systems offline and prevent the Dominion from destroying the 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")