(written from a Production point of view)
A clip show is an episode of a television series that consists primarily of excerpts from previously aired episodes. The clips are generally shown as flashbacks in a new framing story.
Several Star Trek episodes have been clip shows. The Star Trek: The Next Generation Season 2 finale "Shades of Gray" best fits the definition of a clip show. The episode took only three days to shoot, with most of the aired episode comprising clips involving the character of William T. Riker. Production assistant Eric Stillwell was tasked with selecting the clips. "Shades of Gray" was poorly received, with even writer Maurice Hurley calling it "terrible, just terrible". (Star Trek: The Next Generation Companion 2nd ed., p. 94)
In the fourth season, Paramount Pictures asked The Next Generation producers for another clip show to balance the budget for the season. Michael Piller and Rick Berman, however, despised the idea as they didn't want a repeat of "Shades of Gray". Piller commented, "Rick and I discussed it and we both hate, hate, clip shows. We think they're insulting to the audience. They tune in and then you create this false jeopardy and then flashback as their memory goes back to the wonderful time they had before they got trapped in the elevator and all that bullshit." They persuaded the studio to allow them to produce an episode that would be equally under budget but would have some integrity. This became "The Drumhead", a bottle show. (Captains' Logs: The Unauthorized Complete Trek Voyages, p. 220)
Prior to this, the Star Trek: The Original Series episodes "The Menagerie, Part I" and "The Menagerie, Part II" utilized most of the footage from the unaired pilot, "The Cage", in a new framing story. (Captains' Logs: The Unauthorized Complete Trek Voyages, p. 33)
Clips would later feature in Act 8 of the Star Trek: Deep Space Nine finale, "What You Leave Behind", where several main characters reminisced about previous events on the show. It was David Weddle and Bradley Thompson who went back through the past episodes to find clips for the montage. They chose almost ninety clips which they felt best illustrated what the montage scenes were trying to achieve, and the editors whittled them down to a more manageable number. (Star Trek: Deep Space Nine Companion, p. 713)
A variant of the clip show is DS9: "Trials and Tribble-ations", which used not only clips from TOS: "The Trouble with Tribbles" but also new footage integrated together with the original series footage.