As defined by the man who coined the term:
"An Avid fart is an annoying editing technique that involves flashing through a bunch of frames or sped up shots, usually accompanied by a "whoosh" or "thoooom" or "choooonk" sound effect. I believe it was originally used for things like in that Lance Henriksen TV show Millennium when he had a psychic vision. But then they started using it just as a meaningless transition between scenes in many movies and TV shows, because they think it adds production value or style. But it really just makes it seem shitty.
I call it the Avid fart because Avid is the name of the first commonly used digital editing system, and it's this technology that has made Avid farts easy. So editors with no self control start pushing buttons thinking they are adding to the movie by throwing in all kinds of annoying crap that does not add to the movie."
The films of Tony Scott, especially 'Man On Fire' and 'Domino' are filled with great examples of Avid-Farts.