A chaperone was a person who accompanied another person to prevent inappropriate behavior.
In 2372, after Jadzia Dax asked him to accompany them, Julian Bashir surmised (correctly) that she requested his presence to dispel the awkwardness of her meeting with Lenara Kahn (both symbionts having been previously married via Torias Dax and Nilani Kahn) (DS9: "Rejoined")
In 2375, Seven of Nine experienced multiple personalities due to the effects of the Vinculum, one of whom was the son of K'vok, who tried to mate with B'Elanna Torres. Later, when trying to take the Vinculum offline, Seven assured Torres that the Son of K'Vok would not manifest in her again. Torres quipped, "Does this qualify as our second date?", but The Doctor, also present in USS Voyager's engineering to monitor Seven, told Torres to "think of me as your chaperone". (VOY: "Infinite Regress")
In 2376, Tom Paris volunteered to act as chaperone for Harry Kim and the holographic character Maggie O'Halloran. Kim declined, giving Paris' earlier gaff (asking the computer to replace a tire on his automobile in full hearing of Seamus) as his reason. This prompted Paris to follow them anyway and turn Maggie into a cow as a prank. (VOY: "Spirit Folk")