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:
Reformulated in the Net.Legends FAQs "Usenet Rule #4":
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 Nazi Germany or its policies, but rather describes the logical fallacy of Hitler/Nazi comparisons. The most common forms of this are "The Nazis supported X, therefore X is bad/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 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.
Occasionally Josef 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. The Greek political party Golden Dawn has also been used for the same purpose, though it is still mostly used in European parliaments and it is very likely that the only reason why is because only they know what it means. Some people will be topical and use terrorism or slavery 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. A good example of this can be found in one of the debates between Bill O'Reilly and Richard Dawkins. O'Reilly makes the argument that society needs religion to be moral because Stalin, Pol Pot, and Mao were all atheists (this is pretty common in many atheist vs. theist debates-it could almost be a trope of its own, though here it is part of Hollywood Atheist).
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 Adolph 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". There's also a very catchy tune on the subject, as there is with most things.
This trope is (perhaps not surprisingly, given human nature) Older Than They Think. Prior to World War II, the go-to villains were generally Biblical, such as the Nepharious Pharaoh (often thought to be Ramses II) from the Book of Exodus, and Pontius Pilate and Judas Iscariot from the New Testament.
Showcased on the "Germany" episode of 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.
The ''Internetiquette'' short of the fifth episode of the Flemish investigative journalism series Basta / neveneffecten, also showcases those 2 laws ("After a comparison with the Nazis there will always be a stupid guy that will refer to Godwin's law" is the second one) together with plenty of other laws that are present on the internet when they deconstructed the inner workings of an internet discussion. The thing is however as of yet only available in Dutch.
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. And of course, there are neo-Nazis and other such groups that use Nazi symbolism, make racist arguments, and openly compare themselves to the Third Reich, in which case the comparison can probably be conceded as a fair one.
Mike Godwin himself has actually penned articles in major news outlets in attempts to clarify when comparisons to Nazism are and/or are not appropriate in current political debate, and lamented the need for him to write those articles as well.