Although portable, hand-held communicators were available, advanced intercoms were the primary shipboard interface to person-to-person communications systems on the first starships in Starfleet vessels and remained so through the founding of the UFP and until the 24th century.
Intercoms were capable of being tied together for uniform ship-wide output. Both Earth Starfleet and Federation ships possessed this "public address" facility. Starship captains would initiate intraship communications by ordering a bridge officer (typically the communications officer, but also the helmsman or the navigator), to open a channel. (TOS: "The Cage", "Where No Man Has Gone Before", "The Corbomite Maneuver", et al.)
Typically, intercoms were built into walls and bulkheads or were mounted in workstations and desks. Intercoms could be voice-only or, as they developed into the 23rd century, include both voice data and visual images of the speaker. (TOS: "Journey to Babel", "Assignment: Earth"; TAS: "Yesteryear", and others) The panels also included a red button which sounded red alert when necessary. (TOS: "The Naked Time", "Balance of Terror", "Dagger of the Mind", et al.)
By the 24th century, wearable combadges had displaced the intercom as the main shipboard communications device, although the companels are still available as a backup communication system. Intercoms were ill-remembered by 24th century natives who traveled back in time to ships still so equipped. (DS9: "Trials and Tribble-ations")
Background information Edit
The Starfleet intercoms regularly featured on Star Trek: Enterprise were actually non-operational "dummy" units. They could be stuck to the walls of the Enterprise sets wherever needed and were affixed to the walls by double-stick tape. The shiny "speaker" part of each unit was of the same reflective plastic as had been used, in a more "psychedelic" fashion, to make the back wall of the transporter aboard the USS Enterprise, as seen on Star Trek: The Original Series. (ENT: "Stigma" text commentary, ENT Season 2 DVD)