In this story, the main character, Eckles, and some other men go way back into the past to go hunting for a Tyrannosarus Rex. They have to be extremely careful so as not to do anything that will change history. For example, they can only walk on a given path and they have to only kill a dinosaur that they know is going to die in a moment anyway.
But Eckles panics and runs off the path. He apparently steps on a butterfly when he does this. This causes enough changes in history that, when they get back, the US has been taken over by a fascist government.
So small cause = dead butterful; big change = America becoming fascist.