First coined by The Nostalgia Chick in the "Ferngully" episode of The Nostalgia Critic, in which they did a duo review.
A Big Lipped Alligator Moment refers to "the big lipped alligator scene from All Dogs Go To Heaven. This is named after the random musical number sung by a big lipped alligator towards the end of the film; a scene that comes right-the-fuck out of nowhere, has little to no bearing whatsoever in the plot, is way over the top in terms of ridiculousness (even in the context of the movie), and after it happens, no one ever speaks of it again."
In short: It refers to any and all random musical scenes that have no merit in terms of plot, are very over the top and silly, and are oddly never explained or mentioned ever.
Such scenes include:
-The dancing fire gang from Labyrinth
-The pink elephants from Dumbo
-The creepy tunnel scene from Willie Wonka and the Chocolate Factory
-And the sexually undertoned blue Gila Monster (the type of lizard is debatable) song, "If I'm Gonna Eat Somebody (It Might as Well Be You)", from Ferngully
If you are watching a film, and a random musical scene plays, after which you are forced to drop your jaw and go "What the hell was that?", then you have just witnessed a Big Lipped Alligator Moment.