Insanely, randomly, and stupidly smashing down buttons on the game controller hoping that some deity will take pity and let your character perform an actual move. This is usually done by panic-manniquins, idiots, or those who have lost all hope.
Button mashing is most common during fighting games crappily programmed American games. For instance: button mashing in Tony Hawk Pro Skater will get you left in the dust. Button Mashing in Killer Instinct and MK3 can easily get you a win.