This explanation is by no means meant to be totally inclusive of everything that defines Nazi Germany, but just to give an overview of what it is and the reasons it's important here on TV Tropes (mainly because of Godwin's Law).
Nazi Germany is the name commonly used to refer to the German nation when it was ruled by Adolf Hitler in the years 1933-1945, also known as the Third Reich.note Hitler styled "his" new Germany as the "1000-year Reich," for which he got a lot of mockery after the war when it turned out to last only twelve years - although its crimes may well be remembered for over 1000 years. (the First being the Holy Roman Empire,note Which actually kinda-sorta did make it to 1,000 years, although you have to twist the definition of the HRE to include Charlemagne (crowned "Emperor of the Romans" in 800) even though the first "Holy Roman Emperor" was Otto I (crowned in 962), and forget that the HRE spent the last 300 years or so of its existence as a powerless shell ("neither holy, nor Roman, nor an Empire") kept around as a trophy by the Austrian monarchs. and the Second being Imperial Germany under the Prussian Hohenzollern dynasty). The Nazis did not actually change the official name of the country until 1942; for ten of their twelve years in power, it remained simply Deutsches Reich (literally, "German Realm"), as it had been since 1871. In 1942, it was officially renamed Großdeutsches Reich (literally, "Greater German Realm"), to reflect their large territorial gains and the "unification" of the German-speaking lands, and to hark back to the Großdeutsche Lösung ("Greater German Solution") to the "German Question". Therefore, the names "Nazi Germany" and "Third Reich" are conventions for historians, not official denominationsnote When a Modern German talks about the 1933-1945 period, he or she may often call it Nazizeit - "time of the Nazis".
During this time, Germany's policies were dominated by Adolf Hitler and the Nazi Party.note "Nazi" is the phonetic pronunciation of the first two syllables of the party's full name, Nationalsozialistische Deutsche Arbeiterpartei, "National Socialist German Workers' Party" in English. Then, as now, this was seen as an annoyingly long party name. It was intended to sound vague and all-inclusive by combining key words from across the political spectrum (successfully so, by the way - considering its electoral makeup at least in the 1930s, it was the first genuine catch all-party in Germany). But to a German living at the time, the name would have sounded like a bunch of people calling themselves "right-wing left-wingers".* It was also a bit of a cruel joke; the shortening was partly by analogy with Sozi, the then-popular abbreviation for the Sozialdemokratische Partei Deutschlands, i.e. the Social Democratic Party of Germany — who were and remain your standard nonviolent center-left party. They are still around and are one of the major German parties, but hardly anyone calls them "Sozis" anymore. The party and the country it ruled are best remembered for being fanatically racist (even by the standards of the 1930s, their racism was off the charts). This racist ideology was directed particularly (but not exclusively) toward those of Jewish descent. Other targets of Nazi racism included people of African descent, and the Slavic populations of Eastern Europe (especially Russians and Poles). The ultimate goal of the Nazis was the complete extermination of these "inferior races"; in the case of Eastern Europe, the plan was to replace them with German settlers. Though this is by no means the only policy adopted by Nazi Germany, it was so prominent that it has become one of their defining characteristics. The belief in the superiority of what Hitler called the Aryan race would ultimately culminate in The Holocaust — the deliberate, industrialized mass murder of millions of ethnic minorities, dissenters, political opponents of Nazism (especially communists), social "undesirables" and "inferior races" as defined by the Nazi Party; a period of horror and monstrosity that would claim the lives of at least eleven million people, six million of which were Jews, and would give rise to the word "genocide."
Nazi Germany also pursued aggressive territorial expansion. Hitler believed that Nazi Germany and the Aryan race should be the rightful rulers over not just all the German-speaking peoples, or even the Germanic states, but all of Europe. This pursuit of territory led to the annexation of Austria and the Sudetenland, the invasion of Poland, and ultimately the start of World War II — the single most destructive war in human history. This war claimed the lives of an additional 50 or 60 million people, the majority of whom were civilians killed by the Nazi German armed forces.
The extent of the Nazis' desire for conquest is subject to some Alternative Character Interpretation, as certain historians - most infamously A.J.P. Taylor - assert that Hitler's foreign policy was opportunistic and the incompetence of his enemies was as responsible for German gains and the war as himself, while others even believe that Hitler had a Stufenplan (step-by-step plan) for total world domination that would have ended in a global showdown, with the United States on one side and Greater Germany, Italy, Japan and the BritishEmpire on the other. This latter view was most avidly proposed by the now heavily discredited historian Andreas Hilgruber. Since Hilgruber first proposed it in the 1960s, the Stufenplan belief has largely died down, although Taylor's assertions that the annexations of Austria and Czechoslovakia were as much to do with diplomatic blunders as Nazi foreign policy has yet to catch on.
Because of Nazi Germany's involvement in World War II and the unambiguous "evil-ness" of their racial policies, they make convenient enemies in fiction and Video Games. Unfortunately, this has also resulted in people (at both ends of the political spectrum) using the term "Nazi" as an insult towards anyone with whom they may disagree. This practice is strongly discouraged in intelligent debate and is now considered a sign that the name-caller has no actual argument for their case. It's also important to note that just because the Nazis did something doesn't mean that thing is inherently evil purely on account of that association. For example, the Nazis built the first modern highway network (the Autobahn), and it's safe to say that highways are not evil. Probably.
While there are literally thousands of books and other sources of information on Nazi Germany and the historical consequences, here are some brief points of interest:
Nazi Germany was heavily into "racial purity", believing in the superiority of the Aryan race, despite that:
"Aryan" was a synonym for Indo-Europeans, who originated from just over the Caspian Sea, though today the term is usually used in reference to the specific group (Indo-Aryans or Indo-Iranians) who inhabit parts of India and most of Iran (which means "home of the Aryans"). The Nazis believed that the Aryans (Indo-Europeans) were Nordics whose homeland was Iceland or somewhere else in Northern Europe, although some Nazi Occultists like Himmler proposed more zany origin stories, like that the Aryans had been created by the Norse Gods and preserved in ice.
Hitler, despite having blue eyes, wasn't "Aryan" in the slightest. The ideal body image for Germans was supposed to be tall and athletic with blue eyes and blond hair. Ironically, perhaps the three most enthusiastic proponents of "Aryan" racial superiority were Adolf Hitler (of medium height and with dark brown, almost black hair), Heinrich Himmler (tall and stick-like thin, thinning black hair, hazel eyes, glasses) and Joseph Goebbels (very short, black-haired, brown eyes and had a limp). (There was a joke about the Aryan being blonde like Hitler, slim like Goering, and tall like Goebbels.) The only well-known Nazi who actually lived up to the ideal was Reinhard Heydrich, who was super tall, had platinum blond hair, blue eyes, was a champion athlete and a renowned violinist. However, according to Himmler's doctor, Felix Kersten, Heydrich was half Jewish (although this is far from proven). Goering, in his younger years, could have fit as well, but years of hedonism and morphine addiction took a heavy toll on both his body and mind.
Some of the lesser-known facts on the place:
It was the world's first country to run regular TV services.
It was the first country to engage in a major anti-smoking campaign.
It was the first country to develop turbojet aircrafts.
It was the first country to develop ballistic missiles and use them on enemy cities.
As mentioned before, it was the first country to develop an interstate highway system.
Himmler even pointed out, in his 1943 Poznan speech, that "we Germans, who are the only people in the world to have a decent attitude to animals, will also adopt a decent attitude to these human animals". (That last part referred to the Poles and Russians).
The Nazis also enacted conservation laws to preserve German forests.
While homosexuality and interracial relationships were strictly outlawed, the attitude towards straight sex between "Aryans" was quite liberal in the Third Reich, and Aryans were encouraged to have lots of sex and have many children, within marriages or not. Women received welfare from the state to support them if needed, or in rare cases, the children were seized by the SS and raised by them if proven to be genetically Aryan (The "Lebensborn" project). Nude bathing was legalized in 1942. This may have been less about liberated morals (they did ban contraception and abortion for Aryans, except in cases of euthanasia) and more about filling the vast swaths of eastern Europe the Nazis intended to capture and depopulate. The Nazis preached "Aryan" fertility and their idea of the woman's place leaned heavily towards "barefoot and pregnant."
Heinrich Himmler ran the "Lebensborn" program, which was a series of houses created to produce more Nordic blood in the overall German population. If the wife of an SS soldier, who was required to be a non-Jewish Nordic, needed help raising their children, she would receive help at these houses. Nordic women could also put their children up for adoption here, providing the child was pure as well. And single women who wanted children of their own could either adopt or get impregnated by one of those strapping young SS men at a Lebensborn house.
It's commonly known that when Himmler led the SS into Poland, he either killed or enslaved Jews, Slavs, Gypsies, and anyone else who wasn't Germanic. However, as eluded to earlier, if a young Slavic child had Nordic features, he or she would be kidnapped and taken to a Lebensborn house to be raised as a German. Although the Norwegians weren't treated nearly as bad as the Polish, the SS also kidnapped many Norwegian children for the same purpose.
Its trains ran on time, although history tends to overstate this. Hitler liked fostering competition among his underlings — even at the expense of efficiency. Towards the end, the only trains running on time were those heading to Auschwitz...
It actually banned Gothic (Blackletter) writing in 1941 when they were declared to be "Jewish letters," in spite of its pop-cultural association with Nazi Propaganda and the image of Nazi Germany as dark, edgy, and Gothic (which was arguably true even after that).
The SS uniform, which serves as Fetish Fuel for women, and Impossibly Cool Clothes for men, was designed by Hugo Boss. Boss was a big supporter of the Nazis, joined the Nazi Party, and even had a picture in his home of himself with Hitler.
It is a common misconception that the Nazis enacted draconian gun control laws, and as such, the Holocaust was able to happen. In actuality, this is not true:
The Weimar Republic (the democratic government after World War I) was forced to enact very strict gun control laws, as part of the disarmament agreement required by the Versailles Treaty.
By 1928, gun ownership was allowed again, but in very strict conditions, as the Social-Democratic government was fully interested to get rid of armed militias, which included the Nazi SA.
In 1938, the Nazis softened these laws, increasing the length of a permit from one year to three years, lowering the age requirement for said permit from 20 to 18 and eliminating long-gun restrictions altogether. The same law forbad Jews from manufacturing or selling firearms, but they did not lose their right to own guns until two days after Kristallnacht. By that point, they were no longer able to hold government office, marry Germans, go to school with Germans, and the Wehrmacht and SS had all but fully mobilized. It was essentially too late.
The Nazis did strip occupied countries of firearms, though this is understandable, as the last thing you want an occupied nation to have is easy access to weapons for La Résistance.
The 1936 Olympics were the first to feature the Olympic torch relay carrying the Olympic flame from Olympia, Greece to the host city (in this case Berlin).
Everyone knows Jesse Owens won the 1936 Olympic gold medals for the 100m sprint, the long jump, the 200m sprint, and the 4x100m relay, but the first blow to the idea of European eugenically-bred super-athletes was the 1912 Olympics. Oddly, Owens himself is quoted as saying "Hitler didn't snub me; it was FDR that snubbed me." Although when you factor in the fact that, at the time, America's racial policies and attitudes weren't a million miles off from those of Nazi Germany, and that Owens never received any kind of official acknowledgement of his achievements until Eisenhower's administration in the 1950s, this ire becomes a bit more understandable.
There was very little to any rationing in the first half of the war. You know how Americans had victory gardens, were encouraged to recycle everything (scrap metal, kitchen grease, etc)? Brits rationed everything, from gasoline to food? Yeah, Germany had none of that up until Stalingrad. To the average German, it wasn't obvious there was even a war on. They lived in Bread and Circuses. The carpet-bombing campaigns over German cities (which were a British initiative to which Americans rallied themselves) were part of a grand strategic plan based exactly on this: bringing the misery of war to a population that had not yet felt it.
It is often brought up that "socialist" was in the Nazi Party's full name. In truth, this gets complicated. The full name of the Party - "National Socialist German Workers' Party" - would have sounded to a German living in the 1920s and 30s something like "Democratic Republican Liberal Conservative Party" would sound to an American living today. "National" and "German" were right-wing conservative catchphrases (the main conservative party at the time was called the German National Party), and "Socialist" and "Workers" were of course left-wing catchphrases. The name was meant to sound vague and all-embracing, to attract as wide an audience as possible. This reflected the Nazis' nationalist ideology: they always insisted that they were the party of ALL Germans, and opposed democracy for being "divisive". The one constant element in Nazi ideology and practice was genocidal racism and brutal dictatorship, which is what they are remembered for. On most other issues - especially economics - they were actually quite hazy and divided. They had both "left-wing" and "right-wing" factions on economic matters, with the "left" (led by Ernst Rohm and the SA) advocating nationalization of industry, while the "right" (led by Hitler, Himmler and the SS) advocated an alliance with the wealthy capitalists and big business. The "right-wing" faction quickly gained the upper hand, and Rohm and the SA were eventually killed or driven into hiding during the Night of Long Knives. In the end, the Nazis came to power as part of an alliance with conservative political forces, and they mostly continued the economic policies of mainstream German conservatism (which included a 60-year-old tradition of limited welfare state measures, going back to Bismarck). The only extreme or radical element of the Nazi economy was a massive expansion of the military.
That said, they did always oppose laissez-faire capitalism, along with the Italian Fascists. Like Italian Fascism they advocated a so-called "third way" or "fusionist" economic system, in between state socialism and laissez-faire capitalism, so it's economy, though not directly owned by the state, was heavily planned and tightly regulated. (While Nazis were fascist, they didn't want to use the term "fascist" because they felt it sounded too Italian. Essentially, Hitler wanted to establish his own German blend of fascism). Price and wage controls were placed on the economy, industries cartelized, the work force strictly controlled in the government Labor Front when unions were banned, with Four-Year Plans to rival the Five-Year Plans of the Soviet Union. So many regulatory hoops had to be gone through that it severely hampered production. Also, the Nazis in fact never ended unemployment-even with military service and people working in munitions factories, they still had it, but changed the way unemployment statistics were calculated to make it appear better. They also desired to establish autarky (economic self-reliance that did not require foreign trade) but this was impossible since Germany lacked certain essential resources for an industrial economy. This gave them a practical reason, along with gaining living space to contain an expanding German population, for invading and looting other countries of their resources. Author Guenter Reiman named it "the vampire economy" because of this in his book of the same name. So in short: not exactly Socialist and Left Wing, not exactly Capitalist and Right Wing.
...none of which was particularly new in Germany. A regulated-but-still-profit-driven market economy with a heavy military element had been the standard policy of German monarchists and conservatives going all the way back to the 1870s. In Germany, and continental Europe in general, there was no association between right-wing politics and strict laissez-faire capitalism, as there is in America today. In fact, state control of industry (as long as the state was run by the aristocratic or military elite) was seen as a bulwark against socialism, to the extent that it could be used to break unions and other workers' organizations.
The historian Joachim Fest basically says that the Nazis consisted of these outcasts of German society, the massive demographic of angry lower-class men who had no direction in life after WW1 and the Depression.
The important thing about them is that they were overwhelmingly recent arrivals into the lower class. These were not people who had been lower-class all their lives, but people who were used to being (relatively) well-off and respected, who had lost their businesses or otherwise fallen into the lower class as a result of the Depression. So they were angry for being "robbed of their rightful place" in the high-status parts of society, and wanted to get back up there, to form a new ruling elite. As it turned out, they were forced to accept an uncomfortable compromise with the old elite, in order to destroy their mutual enemies (Jews, communists, unions, and of course those pesky French and subhuman Russians).
There were at least two distinctive periods in the 12 years of the Nazizeit ("Nazi Times", as modern Germans call them), so different that it can be inferred there were two different Reichs:
The first period, roughly from 1932, before Hitler assumed Chancellorship, to late 1941, had leaned less on the "Nazi" side and more on the "very conservative and militaristic-bureaucratic empire" side. The military had been dominated by the conservative Prussian senior military class, the Weimar Republic civilian bureaucracy held sway over most day to day running of the administration, the cultural system grunted and groaned under the Nazi purges, yet the way of thinking and ordinary life of most people had not changed enough to be directly felt. There were massive NSDAP rallies, public construction projects, industrial development and some things which were aimed to gain popular support like film, public television, paid vacations for the working class and so on. The early campaigns of the war were more or less classic military conquests. All things which were unpleasant enough and soiled with blood, like extermination of the mentally ill, sterilization, mass murder by the Einsatzgruppen (SS death squads), happened somewhere far away from the eyes of the general population. Death camps were not built yet. Battlefield troops were the old drilled regiments raised under the eye of Prussian generals. The Waffen-SS existed just under the guise of a few units armed with leftovers from the arsenals. Official ideology was Germanic, racist, anti-Semitic and nationalist.
The second period, from 1941 to the end, was quite different. It all started when a few things piled together to warrant a reaction from the Führer: the Norwegian Campaign ended with a Pyrrhic naval victory, the Battle of Britain flopped, the Battle of Moscow ended in a bloody draw, America stirred itself itching to do something, 3.5 million Soviet prisoners burdened the Wehrmacht, and Einsatzgruppen were not up to their task. At this point in the war, the Führer changed his tune: there was no longer Germany against everyone, no easy conquest in sight, no quick victory, and no expenses were to be spared. Suddenly he turned into a "pan-European fighting against the deadly threat of Bolshevism" and the previously despised non-Germans were to be drawn to his cause. The Waffen-SS opened itself to recruitment from all available populations (literally all, down to the "non-Aryan" Turkics and Tatars from Central Asia), the political power of the military class was stymied and the top figures of the NSDAP and SS came out under the limelight, armed forces inflated themselves overnight, the research projects which had been shelved "if no usable result is expected earlier than 1 year" in expectancy of a short war were reactivated, and no pity was to be given to the hated Jews, Communists, Slavs (and the list of Hitler's enemies lengthened itself day by day), new mass-murdering solutionswere found and applied for them. This was the time of the Cool Tanks, Cool Planes, cool weapons, secret weapons, berserker tactics, mass murder, violence in the extreme to nearly-cartoonish levels, the time of Sven Hassel books.
Much like with the Soviet Union and East Germany, there were forbidden political jokes which poked fun at the regime. Here are some:
What's the difference between between Christianity and National Socialism? In Christianity, one died for all. In National Socialism, all die for one.
Two Jews are about to be shot. Then the order comes to hang them instead. One turns to the other and says, "You see, they're running out of bullets."
In 1945, one German asked, "What will you do after the war?" The other replied, "I shall take a trip around Greater Germany." The first said, "And what will you do in the afternoon?
Hitler and Göring are standing atop the Berlin radio tower. Hitler says he wants to do something to put a smile on Berliners' faces. So Göring says: "Why don't you jump?"note A woman caught telling this joke was executed. Her husband had died in Stalingrad.
A Jewish man is charged with killing a Nazi officer and eating his brain. "It is impossible," says the man. "Nazis have no brains and Jews don't eat anything which comes from a pig."
Late in the war, one German asked how to distinguish enemy aircraft. "It's very simple," another replied, "British planes are brown, American planes are silver, and German planes are invisible."
It all ended very, very badly. Way, way, way too big to be an Elephant in the Living Room. For a trope-centric discussion, see Those Wacky Nazis, Godwin's Law, Godwin's Law of Time Travel (two different tropes), Ghostapo and Stupid Jetpack Hitler. For a full index of tropes associated with Nazi Germany, see Reichstropen.
If you want to read more about the regime, the most acclaimed modern comprehensive survey of its era is The Third Reich Trilogy by Richard J. Evans which includes the books, The Coming of the Third Reich, The Third Reich in Power and The Third Reich at War, which are also available in audiobook at Audible.com. Other recommended books are the Hitler duology by Ian Kershaw, Hitler: A Study in Tyranny and Hitler & Stalin: Parallel Lives by Alan Bullock, The Rise and Fall of the Third Reich by William L. Shirer (who was Creator/CBS's radio correspondent until 1941)note Though slightly dated now, was at the time so successful that it actually got him off Joe McCarthy's blacklist through sheer force of sales., The Third Reich: A New History by Michael Burleigh, and The Hitler State by Martin Broszat.