These items of Goa'uld design were advanced computers that were located deep inside a mothership where it controlled vital components of the vessel such as the self-destruct and other ship based systems. An operator from the core was capable of running a ship diagnostic in order to determine any faults that were present within the starship.
A powerful mind was capable of taking over the computer core and thus the ship it controlled.
The core was typically located in a specific chamber on its own level where it appeared as a room with a panel present where instructions were capable of being inputted into the computer.
- Presumably, all Goa'uld vessels shared a similar computer core.
When Supreme Commander Thor of the Asgard fleet was captured by Anubis, he had an implant placed in his brain that formed a link between his mind and the Ha'tak's ship computer. The information was then downloaded into the memory banks of the ship. (SG1: "Revelations")
However, Thor's mind proved to be quite powerful and took over the vessel's systems forcing Anubis to abandon the ship after setting the self-destruct code. An attempt was made to retake the computer core but this failed and the self-destruct was stalled by Thor as the ship approached Earth.