Family entertainment was any kind of recreation suitable for people of all ages.
In 2369, inspired by the appearance of various fictional characters on Deep Space 9, Quark felt family entertainment was the future of profit, envisioning it as "Little holo-creatures running around, rides and games for the kiddies, Ferengis in every doorway selling useless souvenirs". Odo, surprised, asked him if he was going to replace all his sexual holo-programs with family entertainment. Quark clarified that he would be expanding it, not replacing it, much to Odo's continued disgust. (DS9: "If Wishes Were Horses")