(covers information from several alternate timelines)
A combat information center (CIC) was an auxiliary control area used to direct combat operations on military aircraft carriers, starships, and other facilities.
When the USS Enterprise was docked at Alameda, California in 1986, and Pavel Chekov was arrested on board as a suspected Russian spy, the command duty officer in the CIC, Commander Rogerson, took action by calling local FBI authorities after Chekov was arrest. (Star Trek IV: The Voyage Home)
In an alternate timeline, the USS Enterprise-D had a CIC while fighting large-scale military conflicts with the Klingons. When the Enterprise-D encountered the USS Enterprise-C near a temporal rift in 2366, Ensign Thomas was paged over the ship's intercom to report to the CIC. (TNG: "Yesterday's Enterprise")
A combat information center is a real concept used by the US Navy. Unlike Starfleet vessels, the modern Navy rarely staffs both the captain and executive officer on the bridge simultaneously, for reasons of safety for the chain of command. One officer usually mans a post at the alternate location of the CIC.
Starfleet officers C. Alapag, Daviomeisi, and Salter were assigned to the Combat Information Center on Earth in 2372. (DS9: "Paradise Lost")
Daviomeisi and Salter appeared on a personnel assignment log Admiral Leyton was viewing.