Eavesdropping was the act of secretly listening to a conversation or communication. Espionage was a crime based on the act of eavesdropping.
In some cases eavesdropping devices would be used to conduct act. Such devices were utilized by the Ferengi, Makull's species, and Odo aboard Deep Space 9. (DS9: "Crossfire", "The Assignment"; VOY: "Time and Again")
According to Samuel Clemens, after his cigar smoke gave away that he was listening to 24th century Data explain to 19th century Guinan the present situation with his and the Devidians visit from the future, "[e]avesdropping is by no means a proper activity for a gentleman. Nonetheless, the deed is done." (TNG: "Time's Arrow, Part II")
In 2366, while the USS Enterprise-D prepared detailed exospheric charts of the Hayashi system, the USS Hood intercepted the ship at high warp to deliver new top priority orders from Starfleet that required the fastest ship in the fleet and the best people. The orders were not transmitted via subspace communications due to worries that Romulans would be eavesdropping. (TNG: "Tin Man")
After The Doctor admitted to Captain Kathryn Janeway in 2372 that he was eavesdropping on the bridge, he explained that he was "simply trying to monitor issues involving the health of the crew, Captain. When activated, the Emergency Medical program establishes com-links with all key areas of the ship, and I have, on occasion," before stating, "I'm a doctor, not a... voyeur. I am programmed to be discreet." Janeway then referred the situation to Commander Chakotay, who would, "at the earliest opportunity", discuss The Doctor's operating protocols, before ordering "no more eavesdropping." (VOY: "Parturition")