The computer center was a section of a facility that was responsible for the coordination of all information storage and transmission activity.
Federation computer centers[]
Humbolt in the computer center of Starbase 11
In 2267, Commander Spock sneaked into the computer center of Starbase 11 in order to forge a voice transmission to the USS Enterprise as part of a plot to commandeer the ship. Spock was caught by Chief Humbolt, a technician assigned to the computer center, who attempted to stop him, but Spock incapacitated him with a Vulcan nerve pinch. (TOS: "The Menagerie, Part I")
Sometime in the 2270s, Leonard McCoy didn't care for the sickbay of the refit Enterprise, saying it was "like working in a damned computer center." (Star Trek: The Motion Picture)
In the final draft script of "The Menagerie", Starbase 11's computer center was described as "a complex communications and library-computer center, the automated heart of the Star Base."
The computer center as seen in "The Menagerie" was a re-dress of the Enterprise engineering set.