Enterprise, This is Kirk.

Spock here Captain.

Bridge here captain.

Transporter room ready to beam up.

Bridge, this is the captain

Enterprise to Mr Spock.

Captain, Shall I beam downed an armed party?

Scotty here, captain

A communicator, or personal receiver, was a hand-held communication device used by many species for person-to-person, person-to-ship, inter-ship communications. Communicators usually transmitted on subspace frequencies. 

Communicators were used by Starfleet landing parties and away teams; occasionally, communicators were used in situations where normal intra-ship communications were inaccessible (or inadvisable), during the 22nd and 23rd centuries. (TOS: "Mirror, Mirror", et al.) 

Employing a flip-top design, a member of Starfleet spoke directly into the device to give commands and speak with other personnel. (TOS: "The Cage", et al.) 

In some cases, these communicators served purposes beyond basic communication. For example, communicators were often used to allow transporter locks for beaming, thus acting as homing transponders. When used in tandem, two communicators could produce a sonic disruption by using sound beams to create a sympathetic vibration in an unstable object, such as a cliff face. Kirk and Spock used such a technique to ward off a party of angry Capellans on Capella IV in 2267. (TOS: "Friday's Child") 

By the 24th century, these communication devices were integrated into the standard uniform badges and became known as combadges. Users no longer had to speak directly into the communicator, but rather could activate and deactivate them with the touch of a finger. The combadge came into general use by 2344. (TNG: "Yesterday's Enterprise") Late-24th century communicators were composed of a crystalline composite of silicon, beryllium, carbon-70, and gold. (TNG: "Time's Arrow", "The Last Outpost") 

The combadges were also designed to self-activate after their casing was destroyed to aid in the rescue of victims. (VOY: "Time and Again") 

When accessing a computer, the communicator of the person at the terminal logged in the user information, thereby making it possible to trace back who accessed the computer. (TNG: "The Drumhead")