The Third Reich
The Third Reich was a flowery term for the Großdeutsches Reich– commonly known as Nazi Germany. “What if the Nazis won WWII?” is a common point of divergence for AH, possibly because of the horror factor, or because of how much historians know about the Third Reich and its intricate society. The Third Reich is certainly popular with AH publishers, who know that a swastika on the cover of a book will draw the eye of the reader. Good places for swastika flags include flying over Big Ben (the Nazis almost invaded Britain in 1940), flying over the White House (the invasion of America would have taken place several years or some decades after Britain and Russia had been dealt with), and flying on the moon (It has been speculated that the Third Reich would have colonized the moon by 2007).
Why the Nazis could have won:
The Wehrmacht's blitzkrieg strategy conquered several countries and could have reached more.
The Nazis had an efficient chain of command.
Hitler's ambition was to conquer the world, and there were several plans for invading England and other countries that were, in OTL, unrealized.
Why the Nazis lost:
Failure to develop atomic weapons *link
Going to war against Britain without a navy capable of carrying out an invasion *link
The Nazi economy was not as good as people might think. Unlike Britain and the Soviet Union, Nazi Germany failed to institute a proper wartime economy.
Nazi racism meant that they failed to take advantage of people who could have helped them win the war. For example, Jews and Ukrainians. And Nazi ruthlessness turned people against them who might otherwise have been sympathetic to their cause.
Hitler came to believe that he was a man of destiny whose plans were bound to succeed, and so made many bad decisions.
Luck. Britain and the Soviet Union both considered sueing for peace - the British after Dunkirk, the Soviets after the Nazis reached Moscow - but didn't. The Battle of Stalingrad could have gone either way, but in OTL the Nazis lost.
What might have happened if they'd won:
More wars. During the war, Hitler commissioned plans for the invasion of the United Kingdom (Operation Sea Lion
), Ireland (Case Green), Iceland (Operation Icarus), Switzerland (Operation Tannenbaum), Sweden (Operation Polar Fox), Portugal (Operation Isabella), Gibraltar and the Azores (Operation Felix), Turkey (Operation Gertrude), and the Middle East (Plan Orient), not to mention that there would be bound to be partisan movements in the East that would need mopping up. Plans were also made for a big colonial empire in Central Africa, and for the development of long-range bombers and missiles in order to carry out attacks against America. In fact, even before Pearl Harbour Hitler had believed that sooner or later Germany would have to go to war against the United States. He even made some passing references that the Germans would have to overcome even their ally Japan somewhere in the far future, leaving Germany as the sole dominant power over the entire planet.
Ethnic “cleansings”. The SS was to be in charge of the newly conquered lands in Eastern Europe. The Slav populations would be either enslaved, deported to Siberia or exterminated, and their lands resettled by ethnic Germans. Poles were already being enslaved, deported and exterminated before the end of the war.
The other Germanics of Europe (Dutch, Scandinavians, Swiss, etc.), deemed to be racially valuable by the Nazis, were meanwhile targeted for Germanization and to eventually be included in a Groβgermanisches Reich (Greater Germanic Reich). Their populations would at the same time also be encouraged to settle the eastern lands alongside the Germans.
Gargantuan engineering projects. The new Germany would have the largest arches and domes, the tallest skyscrapers and the longest bridges. It would have streets a hundred metres wide and double-decker trains running on two parallel tracks.
A comprehensive, cradle-to-grave welfare state for ethnic Germans.
If Hitler dies, all bets are off. During the war his designated successor was first Hess, then Goering, and finally Goebbels and Dönitz. But if there is a power struggle anyone might succeed him and steer the Reich in a new direction. During the lifetime of the Third Reich the trend was for the SS under Himmler and the Nazi Party under Bormann to assume more and more power; Hitler liked the way they thought.
In the long run, socialism, racism, slave labour and war might not lead to a healthy modern economy.
Wikipedia has an article on the Third Reich.
David Bofinger has written an article on alternate Second World Wars.
Eric Krieg has written an article on lessons learned from World War II.
Virtual History edited by Niall Ferguson (1997) has an essay by Michael Burleigh on what the Nazis had planned for the post-war period.
Wenn Hitler den Krieg gewonnen hätte by Ralph Giordano (1989) is a must read for anyone wanting to write a Nazi victory timeline. It's in German.
Timelines involving a more successful Third Reich
Reich-2 (Halifax becomes PM and makes peace) and Reich-5 (FDR is assassinated in 1933) from GURPS Infinite Worlds. And several other Reich TLs mentioned in the same book.
AH fiction about a more successful Third Reich
The Children's War (2001) and A Change of Regime (2004) by J. N. Stroyar (Hitler invests in atomic research instead of invading the Soviet Union).
Collaborator by Murray Davies (2003) (Germany invades Britain in May 1940 using the French fleet).
In the Presence of Mine Enemies by Harry Turtledove (2003) (America doesn't enter WWII).
The Separation by Christopher Priest (2002) (Rudolf Hess launches a successful coup against Hitler).
Fatherland by Robert Harris (1992) (Germany successfully invades the Caucasus).
The Last Article by Harry Turtledove (1988) (Britain surrenders to Germany in 1941). Short story.
Moon of Ice by Brad Linaweaver (1988) (Germany invents the atom bomb).
SS-GB by Len Deighton (1979) (Germany wins the Battle of Britain).
Two Dooms by Cyril Kornbluth (1958) (The US does not invent the atom bomb). Short story.
The Ultimate Solution by Eric Norden (1973) (Germany and Japan triumph in WWII).
The Man in the High Castle by Philip K. Dick (1962) (FDR is assassinated in 1933; the Allies are completely defeated in World War II)
"Everything's Better With Nazis"
The fact that the Nazis are universally condemned means that they are often a lazy choice of villain in TV and movies, even if the setting is in the present (surviving neo-Nazi cells) or the past (time travel!). This even crops up in AH - AH series featuring crosstime travel between many timelines will always include at least a throwaway mention of a Nazi-victory timeline.
Arguably, the use of the Nazis in kids' TV where nobody ever dies and similar media have undermined understanding of the horrors they actually committed. The same can be said about the German restriction on the display of swastikas, which means that German historical films and computer games always avoid showing Nazi symbols. In some cases, e.g. when WW2 games show Germany as having the imperial or Weimar flag, this can produce unfortunate implications - suggesting historical revisionism.