(covers information from several alternate timelines)
An early radio, ca. 1930
Radio was the name of a 20th century device which was used to transmit or receive sound or signals by means of electromagnetic waves. This name was also used to describe signals which were transmitted on a specific bandwidth. Methods of radio transmission included amplitude modulation and frequency modulation.
Uses
In 1930 Captain James T. Kirk (traveled back in time from 2267) and Edith Keeler passed by a radio repair shop on the streets of New York City, where a radio was playing the song Goodnight, Sweetheart, Goodnight. (TOS: "The City on the Edge of Forever")
In an alternate timeline, communications officer Hoshi Sato analyzed radio signals from the planet Earth during 1944. Among the transmissions was a speech by Winston Churchill and a broadcast of The Shadow. (ENT: "Storm Front")
In 1957 a group of Vulcan scientists including T'Mir observed Earth radio broadcasts concerning the launch of satellite Sputnik I. (ENT: "Carbon Creek")
Later that year, when T'Mir and Mestral were stranded in Carbon Creek, they overheard part of a baseball game on the radio. (ENT: "Carbon Creek")
In 1967 a young Henry Starling listened to rock and roll music on his portable radio during a camping trip in the High Sierras. (VOY: "Future's End")
When the USS Enterprise (from 2267) accidentally traveled back in time to 1969, the crew received a radio news broadcast about the first manned Moon shot scheduled for that Wednesday. (TOS: "Tomorrow is Yesterday")
During another time travel incident, the USS Enterprise (from 2268) traveled back to 1968 in a mission to study that period of Earth history by listening to radio and other broadcast communications. (TOS: "Assignment: Earth")
In 1986 the pair of Humpback whales known as George and Gracie were equipped with a special radio transmitter with a frequency of 401 megahertz when they were flown to the Bering Sea off of Alaska. (Star Trek IV: The Voyage Home)
In 1996 the SETI project used radio broadcasted signals to contact possible alien lifeforms. (VOY: "Future's End", "Future's End, Part II")
In 2168 the Horizon used conventional radio to send out a distress call soon after visiting the planet Sigma Iotia II, which resulted in the transmission being intercepted a hundred years later. (TOS: "A Piece of the Action")
Subspace radio was an advanced radio system. (TOS: "The Cage")
In 2266 the USS Enterprise received a radio transmitted distress signal from a planet identical to Earth. (TOS: "Miri")
In 2267 the USS Enterprise found the SS Botany Bay floating in space, transmitting an old radio signal. (TOS: "Space Seed")
In 2268, Ekosian Nazi Party captive Isak asked fellow captives James Kirk and Spock if they had a radio when he observed them removing rubidium crystals from their arms. Later that year, Lieutenant Uhura monitored several radio broadcasts on the planet 892-IV. (TOS: "Patterns of Force", "Bread and Circuses")
Radio was a popular broadcasting medium on Sigma Iotia II in the same year, featuring music and commercials. Commander Spock also used a local radio frequency to contact the USS Enterprise. (TOS: "A Piece of the Action")
While star charting on the outer fringe of the Milky Way Galaxy in 2269, the Enterprise discovered a source of mysterious radio emissions originating from an insectoid pod ship trapped in orbit of the dead star Questar M-17. (TAS: "Beyond the Farthest Star")
The V'ger entity used late-20th century radio signals to call it's "Creator" while approaching Earth in the mid-2270s. (Star Trek: The Motion Picture)
Fictional detective Dixon Hill had a radio set in his office in the holonovel adaptations based on the series of novels featuring the character. (TNG: "Manhunt")
In 2367 the USS Enterprise-D studied radio broadcasts of the planet Malcor III in preparation for a possible first contact. (TNG: "First Contact")
A spaceborn race existed in the 24th century which communicated through radio signals. (TNG: "Galaxy's Child")
Amelia Earhart's airplane signaled an "SOS" distress call on AM radio from a Class L planet in the Delta Quadrant in 2371. The signal was intercepted by the radio of a 1936 Ford truck found floating in space by the USS Voyager. (VOY: "The 37's")
In 2372 The Doctor and Danara Pel listened to the song "My Prayer" by The Platters on a car radio of a 1957 Chevrolet in the holodeck. (VOY: "Lifesigns")
In a holodeck recreation of Sainte Claire, France during World War II on the USS Voyager, invaded by a group of Hirogen hunters in 2374, the Allied forces used radio news broadcasts to transmit messages to the French Resistance. The Nazis also used radio for communication, which the Resistance had to sabotage. (VOY: "The Killing Game", "The Killing Game, Part II")
In the same year, Tom Paris listened to surf rock music on a portable radio in his Grease Monkey holoprogram while working on a 1969 Chevy Camaro automobile. (VOY: "Vis à Vis")
Still in the same year, in Captain Benjamin Sisko's vision from Prophets of 1953 New York City, Benny Russell had a radio set in his home. (DS9: "Far Beyond the Stars")
In the The Adventures of Captain Proton holonovel series, characters (including Doctor Chaotica) often used radio transmissions as a way of interspace communication. (VOY: "Night", "Thirty Days", "Bride of Chaotica!")
Scientists on Kelemane's planet used radio broadcasts to transmit their messages to the "Sky Ship" (USS Voyager) during one period of the planet's history in 2376. Seven of Nine received their transmissions which led Captain Janeway to send The Doctor to an away mission on the surface. (VOY: "Blink of an Eye")