Consider Die Banalität des Bösen by Hannah Arendt; from her observations at the trial of Eichmann, she drew the conclusion that ideological affinity with Nazism, or even a broader affinity with antisemitism, did not play a big role in the motivations of functionaries to join ranks with - or rather, in - the NSDAP and join the worst of atrocities. Eichmann was a pure careerist and acted in deferrence to what he perceived to be the legitimate state authority.

※ Supplemental: As several readers have commented, Arendt's Eichmann in Jerusalem appears to have been discredited. Please refer to this post by /u/DanKensington, and the critiques linked therein, for further information.

Christopher Browning, in Ordinary Men, analyses the motives of a group of middle-aged police reservists deployed to Poland for the purpose of murdering the local Jewish population in the rear of the eastwards-advancing military in 1942. He comes to much the same conclusion: that few of these men held any ardent National Socialist convictions, and even the older Imperial German brand of antisemitism was rare among them - and yet most of them participated in the worst atrocities. He makes out a few individuals with enough of a moral fiber to back out of the mass murder of children, for example, but says that by and large, peer pressure, careerism, deference to authority were largely responsible for the behaviour of this group.

Adorno, in The Authoritarian Personality, on the other hand, identifies the presence of a segment of the population with a certain psychological profile - the eponymous Authoritarian Personality type - which is characterized by deferring to authority while also seeking it over others. He curiously identifies this to be correlated with antisemitism (though also with discriminatory attitudes towards "negroes"). The research that went into TAP was conducted in the US population during the FDR era; YMMV regarding historical racist social-political order in the United States, but there clearly never was a pronouncedly antisemitic, let alone totalitarian-so, political system in place in the United States, and yet Adorno was able to identify this population subgroup. From this, he concludes that "sympathies with Nazims" pre-cede the advent of actual Nazism - there is a subgroup of the population which sympathizes with such ideas and forms of societal organization, preceding any state-organized indoctrination (Adorno himself, being a Freudian, proposes an oedipal mechanism for its genesis, which, hasn't aged well as an explanation and which, again, YMMV on, but the point that it is not the product of state-side indoctrination stands). When a movement arises under the right historical conditions, it simply galvanizes the support of those who already carry its ideas.

Edit:

/u/sloby has thankfully pointed something out that I think warrants a little more discussion (check their response to this post). There is indeed a subtle difference between the concept of "peer pressure" we are familiar with, and the social dynamic in the group examined in Ordinary Men. We would probably think of peer pressure as something that is produced by communication: the peer group makes clear what its values are, and we are compelled to comply. The social dynamic here was a different one; it was naturally impossible to communicate about whether one agreed with the operations on an ideological or ethical level. What the members of the group were left with, then, was the tacit assumption that everyone else did not object, even when everyone individually might have. We have to think about this kind of of peer pressure as being sustained not by the group, but wholly "internalized" and maintained by silence, by the impossibility to question.

This is also why the only resort for those with the mentioned "moral fiber" was to back out not because of ideological objection, but because of being "not man enough" to conduct the deed. While this was self-effacing, it was not seen as political defiance, but simply a personal shortcoming, and those who stepped out of particular atrocities on such ground were not persecuted in the way political opponents to the Nazi regime were. Browning makes this into the point that, in principle, although it is valid that political object would have been suicidal, this does not mean that those who went along with the atrocities were strictly "forced"; they would have had a way out.

If we are thinking about this question considering contemporary challenges with political radicalism, we must start by mentioning that the Allied Control Council, was deradicalizing a country that had been completely defeated and occupied. The catastrophe that national socialism brought on Germany should discredit it for a lot of people already and was a point that the Allies wanted to "drive home" during their administration of the country:

"It should be brought home to the Germans that Germany’s ruthless warfare and the fanatical Nazi resistance have destroyed the German economy and made chaos and suffering inevitable and that the Germans cannot escape responsibility for what they have brought upon themselves. " - Directive to the Commander in Chief of the U.S. Occupation Forces (JCS 1067) (April 1945)

Surveys from 1945 – 1949 showed a majority of those surveyed in the US zone of occupation believed that “Nazism to have been a good idea but badly applied”. Generally, these sorts of attitudes continued to decline into the 1950s. That’s according to surveys conducted by US authorities that you can look up and Tony Judt uses them when considering denazification in his book “Postwar: a History of Europe since 1945”.But there were difficulties, the allies realised early on that pursuing all members of the Nazi party and government functionaries was impractical for the running of the country (over 3.5 million former Nazi party members in the US zone alone) and risked creating a sizeable group of second-class citizens who through their social exclusion and that of their family, could be a source of future political violence. One way they avoided this was by creating different categories of Nazis:

I. Major Offender; II. Offender; III. Lesser Offender; IV. Follower; and V.Exonerated. (Law for Liberation from National Socialism and Militarism, 1946)

Category IV. (follower) also quite widely known by the German term Mitläufer, is of particular interest, as it covers large swathes of the German society who joined the Nazi party for reasons other than believing in Nazi ideology. That was a large group of former Nazis who didn't need ideological deprogamming but still some form of punishment for their involvement. This divided the group and allowed a large part of them to pretty quickly return to normal lives.

The Allies quickly turned over the policy of denazification to the Germans themselves and the government of the Federal Republic (West Germany) pretty much stopped it immediately. Adenauer himself described denazification as a witch hunt and brought in amnesty legislation that benefitted almost 800,000 people.  Then of course more widely known are how the necessities of the cold war also created a justification for allowing a lot of former SS and Wehrmacht back into important positions. It’s not surprising that only decades later Nazi party members or functionaries are exposed as continuing their career almost uninterrupted in the FRG, particularly notorious is the foreign office and offices of public prosecutors. The latter being the organisation that was tasked with bringing remaining Nazi war criminals to justice, an activity very likely hampered by former Nazis in their own ranks and why only now that the few remaining concentration camp guards are being prosecuted.

There were several right-wing parties that formed post-war made up of former Nazi party and SS members such as the Deutsche Reichspartei. These parties were unsuccessful politically and/or eventually banned by the German constitutional court for being anti-democratic.

Bit of an aside but issue of Vergangheitsbewältigung (dealing with the past) was picked up again by the first post-war generation (68 generation) during the student movements of the60s and really laid bare the failings of denazification, in its aim of removing Nazis from public life. At the time the German Chancellor was Kurt Georg Kiesinger, a former member of the Nazi party who would fit the definition from his own words as a Mitläufer. The 68 generation though is important for ensuring the construction of memorials for victims of the Nazis, public education about Nazi crimes, the development of academic studies of the Holocaust and programmes of compensation for surviving victims of the Nazis. Still here are questions of how effective these things are, Harald Welzer (Collateral Damager of History Education: National Socialism and the Holocaust in Germany Family Memory, 2008) found through interviews, that the younger generation is very knowledgeable on the topic but struggle to accept that it is also intertwined with their own family history, even in situations where their grandparents admit to having supported the Nazis and engaged in war crimes. It also supports the point that racial ideology doesn’t seem to have gone that deep, but support for the Nazis was widespread until 1941/42.   

So far, I’ve mostly mentioned the facts of denazification and how it was ineffective in removing former Nazis from public life. But maybe we’re starting from a false premise? How effective Nazi indoctrination was is debatable. The Nazis themselves at least early on, were selective in what aspects of their ideology they tried to spread, knowing that antisemitism wouldn’t have much appeal in some areas. They also made efforts to hide the crimes committed by the regime, something that one would think unnecessary if most of the population was indoctrinated.  I mentioned the category of followers, it was a widespread phenomenon that people joined the Nazi party to advance their careers, and in the aftermath tried to defend themselves as simply following orders, such as the notorious case of Adolf Eichmann.

This might be another case of even modern audiences falling victim to Nazi propaganda, many actions such as parades, flag-waving and set-piece speeches were orchestrated to create the impression of ideological unity and belief, not the result of it. It’s a strategy used by many authoritarian regimes (Check out One Day We Will Live Without Fear: EverydayLives Under the Soviet Police State, by Mark Harrison). In a society where expressing your opinion can be punished, discontent becomes a private matter and shows of unity can create the impression that everyone else is content in the system and that only the individual is discontent. That’s not to say the Nazis were never popular, but they were popular for their perceived successes with the economy and the success of the initial military campaigns. Popularity that begins to waver by late 1941, early 1942.

Considering what I've typed so far and to directly address the question: We don't know what people truly believed we can only guess based on their actions. It's a fact that a lot of former Nazis and associated functionaries were never made to explicitly renounce Nazi ideology, but they did not seem to act on this ideology if they still held it, which could be considered a success for denazification. Of course, things like racism and homophobia remained a part of German politics post-war but these aren't unique to Nazi ideology.

There is evidence that denazification was successful amongst the general population, while it’s still debatable how deep ideology went amongst the German population to begin with. As it seems a lot of support for the Nazi party was because of their perceived successes in areas such as the economy and security, and not due to the Nazis racial policies.

After the war Nazism no longer existed as a political project and the post-war political consensus committed to preventing the circumstances that had enabled its rise. If we consider denazification as wanting to remove nazi ideology and not just Nazi individuals from German politics, then I would say it was successful and was actually relatively easy for the Allies because the circumstances were in their favour. My last point that Nazism (and fascism in general) became irrelevant and denazification occurred almost automatically is something Hobsbawm put quite succinctly “ …post-war politics soon reverted to exactly what it had been before democracy was abolished in 1933, with the exception of a slight shift to the Left. Fascism disappeared with the world crisis that had allowed it to emerge. It had never been, even in theory, a universal programme or political project.” (The Age of Extremes, p.176)

As a supplement to the answers here, I will link to an earlier answer I wrote on this subject. The depth of support for Nazism and its program after the war actually was an area of interest among the US occupation authorities, and they conducted a number of polls on this subject.

The long and short is the results varied quite a bit whether one talked about favorability ratings for Hitler v the Nazi party v aspects of either's belief systems or goals. There was a core of support for all of these, but among respondents it was never a majority (although it frankly is surprising that so many people would answer in the affirmative to any of this in surveys conducted by a foreign occupation force).

ETA - I will also mention that this phenomenon connects to ideas developed by Ian Kershaw, especially in books like The Hitler Myth: Image and Reality in the Third Reich. Namely that Hitler as a leader was more popular among the Germans than the NSDAP as a party (it's worth noting that even in the March 1933 Reichstag elections, marred by massive violence and intimidation, a majority of voters still cast votes against the Nazis). But even Hitler's popularity among Germans was not unquestioned, and was heavily tied to the successes and failures that Germany experienced. Furthermore, Nazi leaders themselves were very much aware of this discrepancy in support, and they (and Hitler) worked very much to build popular support around the image of Hitler far more than pushing for popular support for all or most of Hitler's ideosyncratic beliefs.

If anyone would be interested in how this process went in Austria (e.g. how it was undermined at every turn by an Allied-sponsored 'first victim' narrative that prevailed there in the post-war period and ultimately produced no end of scandal - see Kurt Waldheim etc.), I could write up a full-length answer if there is sufficient interest. I am currently writing a PhD about the British occupation of Carinthia and Styria. In any case I strongly recommend checking out the work of Robert Knight, whose publications on this topic have done a great deal to annoy the Austrian government