Movies become console verified when they are successfully played back on real hardware.
Since the beginning, tool-assisted speedruns were created such that given super-human abilities, it would be theoretically possible to perform them on actual consoles. Using special hardware (i.e. replay devices), many movies have been shown, with video evidence, to play correctly on console. Such movies can be found below.
Getting these tool-assisted speedruns to playback on a console, is a testament to how accurate the emulators used on this site have become, and that all tricks and glitches depicted are really possible in the real games. Verifying movies to this level has been achieved on the NES, FDS, SNES, N64, Genesis/32X, Gameboy/Color/Advance, and the Atari 2600. For more information on how this is accomplished, read the verification guide. Also check out the forum for discussions on this topic.
Lists of console verified movies per platform: