A playback from a flight recorder showing Spock and McCoy
Playback was a term used for watching or listening to something that had been previously recorded, such as a holo-recording or log entry. This was generally accomplished by pressing play on any media player device, or with voice commands.
In 2285, Admiral James T. Kirk used the USS Enterprise's flight recorder to playback the events that happened in the engine room leading up to Spock's death. (Star Trek III: The Search for Spock)
When it was believed that Aquiel Uhnari was dead, Geordi La Forge had the computer do a playback of her personal logs. (TNG: "Aquiel")
In 2373, when the Kazon-Nistrim captured the USS Voyager, leaving her crew marooned on an alien planet; the Doctor recorded a log entry for Ensign Lon Suder to find and playback, so as to assist him in taking back the ship. (VOY: "Basics, Part II")
That same year, while practicing his opera singing skills with a holographic "diva;" at multiple points during their singing he paused the musical playback. (VOY: "The Swarm")
When the playback of a Tribbles cereal commercial began, the video tape tracking auto-adjusted itself. (ST: "The Trouble with Edward")
In a malfunctioning holoprogram, Trip Tucker told the computer to pause the playback of Neelix going to bed in his quarters on the USS Voyager. (VST: "Holograms All the Way Down")