Radio silence, or subspace silence, was an operating protocol for starships that were either in combat or operating in a combat zone. When radio silence was invoked, the ship sent absolutely no outgoing communications – not even to acknowledge incoming transmissions.
In 1986, while on a mission to obtain humpback whales to save Earth of the future, James T. Kirk had his crew divided into three teams. The teams were given their assignments and each individual was given a phaser and a communicator. Kirk instructed his crew that, save for emergencies, that they were to maintain radio silence. (Star Trek IV: The Voyage Home)
In 2267, after Spock had commandeered the USS Enterprise with forged Starfleet orders, he ordered Lieutenant Nyota Uhura to maintain radio silence, when the ship was hailed by Starbase 11. (TOS: "The Menagerie, Part I")
In 2268, when the Enterprise received a priority 1 distress call, subspace silence was maintained until the ship reached Deep Space Station K-7, upon which James T. Kirk, seeing that all was apparently peaceful, broke silence to ask the nature of the emergency. (TOS: "The Trouble with Tribbles")
In 2366, a replica of Captain Jean-Luc Picard enforced a form of radio silence on the USS Enterprise-D, belaying an order from Commander William T. Riker to contact the USS Hood to inform them of a change of orders, as doing so would have revealed that these orders were fictional. He then stated that there would be no further communication off the ship without his prior authorization. (TNG: "Allegiance")
In 2368, when the long-term memories of the crew of the USS Enterprise-D were temporarily erased by the impostor Kieran MacDuff, the false orders from Starfleet placed in the ship's computer stated that they were to maintain radio silence. This was part of the ruse, as any contact with Starfleet would have revealed that the supposed orders to destroy the Lysian Central Command were false. (TNG: "Conundrum")
While it was stranded behind enemy lines in early 2374, the USS Valiant observed strict radio silence to avoid detection by the Jem'Hadar. This had the unfortunate side effect of leaving Starfleet Command unaware of the fact that the Valiant was being operated solely by the surviving cadet members of Red Squad. (DS9: "Valiant")