If a picture's worth a thousand words, then an animated GIF or video must surely be a great way to explain or illustrate a problem or solution, and sometimes it's the best way. Since people sometimes ask how to make these, I thought a question and a variety of answers would be useful.
In an earlier question I suggested that we ask for the ability to embed videos into posts. This is because it adds little to the size of the post, reducing the downloading burden on browsers, but allows a single click or inline viewing. (See this question on Gaming for how it would work. However, this facility hasn't been enabled for the main Mathematica SE site. I think it should be added (feature request tag...:)).
I suppose you should downvote this question if you would prefer to see fewer or no animations, regardless of how useful they might be.
I'm assuming that community votes (and edits) will indicate whether there are too many gratuitous animations in questions or answers!