Also known as "Godwin's Rule of Nazi Analogies". Originated by Richard Sexton, and popularized by Mike Godwin of the Electronic Frontier Foundation (and of the Wikimedia Foundation until 2010) in 1990 in the form:
Appended in the Net.Legends FAQ s "Usenet Rule #4":
"Any off-topic mention of Hitler or Nazis will cause the thread it is mentioned in to come to an irrelevant and off-topic end very soon; every thread on UseNet has a constantly-increasing probability to contain such a mention."
It is generally accepted that whoever is the first to play the "Hitler card" has lost the argument as well as any trace of respect, as having to resort to comparing your adversary to the most infamous mass-murdering dictator in history generally means you've run out of better arguments. Thus, once such a comparison is made, the thread is finished and whoever mentioned the Nazis has automatically lost whatever debate was in progress. This principle is itself frequently referred to as Godwin's Law.
The usage of Godwin's Law also has "Henderson's Law" as a corollary, referring to an observation by Joel Henderson that while Mike Godwin specifically stated this to pertain to "gratuitous Hitler-comparisons", Godwin's Law has been frivolously thrown at any comparison no matter how accurate or on-point. Case example: Jon Stewart of The Daily Show criticizing comparisons to Hitler.
Note that the Law is not supposed to apply to serious discussions of Fascist Germany or its policies, but rather describes the logical fallacy of Hitler/Nazi comparisons. However, Godwin's Law is not intended to apply to on-topic comparisons of neo-fascist groups to Nazis.
In other words, it's OK to call these un-American white supremacists exactly what they are: "By all means, compare these shitheads to Nazis. Again and again. I'm with you," Godwin said on Sunday evening.
The most common forms of Godwin's Law are "The Nazis supported X, therefore X is bad", or alternatively, "The Nazis opposed X, therefore X is good". Whether using "Nazi" as a random insult falls under the Law is a matter of debate. Unfortunately, this has become so popular as to come full-circle, making any discussion of totalitarian regimes susceptible to "HAY GODWIN'S LAW HURR".
As Quirk's Exception points out, attempting to invoke Godwin's Law intentionally in order to force-terminate a thread rarely works. All the same, shouting "Hitler!" is a fun way to express your opinion that a thread should be put to rest. Of course, it's also helping Hitler indirectly, as his greatest expressed wish was to be remembered forever, which means that you're just as bad as Hitler, you horrible Nazi-bastard. However, this corollary is not in the law itself. Likewise, trying to bait your opponent into breaking the law is poor form too. Sometimes commenters will try to get around mentioning Hitler's name directly (e.g. "You know who ELSE got rejected in art school? THAT'S RIGHT."), but this is really no better.
Events in the Harry Potter fandom in the early 2010s have led Fandom_Wank to coin the Pacific Theater Corollary, in which someone invokes the atomic bombing of Hiroshima and Nagasaki in the same way. FandomWank also gave us this excellent antidote to frivolous Nazi comparisons.
Occasionally Stalin is referenced, often by people who are aware of Godwin's Law but want to convey a similar message; in this case, this might slip into the Commie Nazis trope. Some people will be topical and use Osama Bin Laden or slavery (especially America's brand of it) as the canonical ultimate evil. However, any of these can also be seen to have violated Godwin's Law, since the point remains: comparing your argument to a clear and non-debatable atrocity is simply bad debating, since it implies that the opposition has no redeeming qualities whatsoever, and is obviously insensitive to real-life victims and their descendants.
In fan circles, there's the "true fan" argument, in which anyone who plays the "true fan" card automatically loses the argument:
"After a rather influential message by M Sipher in 1997 the term 'true fan' has taken on a whole new meaning among some TransFans. It's a sort of twist on Godwin's Law where anybody who accuses somebody of not being a true fan automatically loses any argument, and is often discounted as a buffoon afterwards."
Note that some lay claim to the ability to invoke "Bentsen's Defense" (i.e. "I knew Mike Godwin, Mike Godwin was a friend of mine -- you, sir are no Mike Godwin") to negate invocations of Godwin's Law. Note, also, that this is only available to those who can truthfully make the statement and that it is designed to foster reasoned discussion — not to deter it.
Pre-Mike Godwin and its prevalence on the internet, the spoken and written word version of this was called reductio ad Hitlerum or argumentum ad Hitlerum, coined by ethical philosopher Leo Strauss in 1953. It means pretty much the same as Godwin's Law: "A view is not refuted by the fact that it happens to have been shared by Hitler."
George Orwell said something similar in his 1946 essay, "Politics and the English Language", where he noted the new definition of fascism had pretty much become "anything you don't agree with".
This trope is (perhaps not surprisingly, given human nature) Older Than They Think. Prior to the 1940s, the go-to villains were generally Biblical, such as the Pharaoh of the Exodus, Pontius Pilate, or Judas.
Showcased on QI as "Godwin's Rule", giving an example of Hitler's love of animals (and disgust of fox hunting) as a fallacious reason to keep fox hunting legal, though this wiki considers that a wholly separate logical fallacy: Hitler Ate Sugar.
Heavily overlaps with Demonization and can be seen as its modern, secular adaptation. See also Abomination Accusation Attack. Not to be confused with Godwin's Law of Time Travel. It might be justified when seriously discussing genocide, since the Holocaust is pretty much the Trope Codifier that most other genocides are measured against. The word was even first coined in reference to it.
Even Nazis aren't immune to it. On neo-Nazi forums, one participant will eventually accuse another participant of being a Jew.