This term has been bandied about quite a bit in recent years, although the sentiments go back quite a bit farther even to the glam rock/metal days. Metal is a genre that is typically better when it is underground. As is the case with all music, when something is popular, it gets watered down and there are more imitators than there are genuinely talented groups with something original to say. Pop music is ultimately disposable and many of its listeners are not hardcore fans of music, they just seek the immediate gratification of a hit song. This is not normally the case with metal. Many people who listen to metal will do so for their entire lives.
Back to false metal. I have seen many different definitions of the term, but here is mine: false metal bands claim to be metal or seek to be affiliated with the metal scene, however their music bears little or no resemblance to metal.
There is a spectrum of course. To meet my definition of metal though, a band must be predominantly metal, that is a majority of their sound is metal. If a band is more hardcore than metal, they are not a metal band.
I do not deny that many false metal bands may actually have some metal influence, however some influence is not enough to actually consider a band metal. Take the case of Korn. Undoubtedly Korn has some metal influences in their music. For example, the breakdown in the song "Ball Tongue" off their debut album is lifted from Morbid Angel's "Angel of Disease". Korn also has very strong rap/hip hop influences as claimed by their bassist Fieldy. No one would suggest however that Korn is a rap/hip hop group, so then why do people try to pass them off as a metal band? Their metal influences are really no greater than their rap influences. Korn is principally influenced by alternative rock and grunge music with some other influences thrown in, and that is therefore what they are.
Slipknot is in much the same position albeit with fewer rap influences and more metal influences. Nevertheless, their grunge/hard rock influences actually dominate their sound. Slipknot has much more in common with Korn than it does with Metallica/Pantera/Cannibal Corpse/whoever the hell else people claim they sound like.
Avenged Sevenfold is another example. This is a band that fits the above definition of a group having more hardcore influences than metal influences. Yet the band claims itself to be a metal band. I have tried several times to listen to them and figure out where their metal influences are and have come to the conclusion that there may be a metal riff in each song, but by and large, metal is completely lacking. Besides the band is ungodly terrible, particularly after their singer M. Shadows lost his voice, his vocals now consist of high-pitched whining. Besides their music, visually the band resembles what would happen if Guns 'n Roses and My Chemical Romance mated. Further destroying the band's metal credibility is their affiliation with Good Charlotte which includes one of the guitarists contributing a solo to a GC song.
All that being said, here is a partial list of bands that fit my definition of false metal:
Guns 'n Roses
I Killed the Prom Queen