A home video was a filmed record of an event or daily life.
In 2268, James T. Kirk showed the children of the Starnes Exploration Party videos of themselves having fun at a picnic with their now-deceased parents in order to convince them to resist Gorgan's will. The tactic was successful. (TOS: "And the Children Shall Lead")
In the fantasy created by Barash, Commander William T. Riker and Jean-Luc Riker watched a home video of the latter's seventh birthday, in which Min Riker appeared. It was Riker's first tip off that something wasn't right about his "future" remembering the holographic character Minuet who looked exactly like Min. (TNG: "Future Imperfect")