More can always be said on this, but this old thread and this old thread both have some useful information from /u/commiespaceinvader, while this one from /u/gingeryid might also be of interest.

Follow up question (apologies if I'm using incorrect/intensive terminology, no offence is intended): Did the Nazis distinguish between "racial" Jews, i.e. people with Jewish heritage whether or not they were practicing and religious Jews who may have converted for marriage or some other purpose but may have had a "pure Aryan" heritage?

It's important to consider/remember the history of anti-semitism in Europe and how on most of the continent Jewish people had been marginalized, vandalized, and domineered since the crusades. They were only really returned to part of society in the late 1800s - not long before WWI and WWII. They were already segregated and struggling in most countries, and before the nazi's came to power in Germany.

The NSDAP (nazi party) attempted and failed to seize power by force in the early 1920s, but would eventually succeed to political dominance by 1933, and when the German president died in 1934 Hitler assimilated the position into his existing role and became the dictator, allowing him full control of the country. The NSDAP was openly anti semitic (boycotting and vandalizing Jewish shops, blocking people from entering) and had passed some legislation (denying Jews work in certain fields, restricting university enrollment) but couldn't fully and unimpededly enforce/enact their beliefs until this point in time.

Throughout all of this the nazi party used propaganda much to their advantage in the written and spoken form to affect public perception. Think of the contemporary "flat-earthers" - we have empirical evidence to definitively prove the earth is round, yet we have a group of people who seemed absolutely normal and rational until they bought into flat-earth propaganda. This is a silly example (and on a much smaller scale, I know) but I'm trying to show how effective propaganda can be in influencing a movement. When it is used masterfully it is a weapon, and as a political tool they weaponized it and wielded it with precision and impunity creating an atmosphere that was susceptible to their rhetoric from the late 20s onward.

They also had the SS ("security" staff), the SA (storm battalion - armed "protection") and from the late 30s onward the gestapo (secret police) organizations at street level enforcing (literally) their beliefs, rhetoric, and laws against the Jewish people. In every way possible - physically, theoretically, economically, legally, and religiously the Jewish people were increasingly persecuted, and steadily backed into a corner.

The Nuremberg Laws of 1935 were the watershed moment. They revoked citizenship from Jewish people, forbade marriages of mixed faiths and race with the punishment of imprisonment, but most importantly the laws defined what it was to be Jewish. They now had a state sponsored definition/label to apply and use at their discretion and will. It's not about trying to tell the difference - they used the law to determine race in a scientific, clinical fashion. Next they took a census with a view to establishing who was/wasn't Jewish based on this definition/criteria. Now put yourself in that situation with the violence, propaganda, and laws in place - do you lie? Do you put your family at risk? Violence, jail time, disappearance and death was commonplace, do you risk even further punishment? Or, do you comply, and survive?

The famous culmination of this work was in 1938 on Kristallnacht (The Night of Broken Glass). A German politician was shot in Paris by a Polish Jew - the son of parents who had been deported from Germany to Poland (his alleged motive). As a result the minister of propaganda Joseph Goebbels called for retaliation against the Jews - a people already on the fringes of society, easily identifiable, who have no access to social welfare, or protection of any kind. The nazis set about destroying as many homes, businesses, and synagogues as possible, and openly attacked Jewish people in the street in retaliation. As a race the Jews were then issued a massive fine as reparation, and had a portion of their possessions seized by the state (real estate, shops, homes, etc.).

The forced wearing of the badge also came about in 1938 and had similar intentions and punishments as the Nuremberg Laws. Once again put yourself in the shoes of a Jewish person in that time/space. You are not a considered a citizen of the country, you are lucky if you still have your home, your health, your place of business, or your "freedom". You don't know what's going to happen if you comply, but you know exactly what will happen if you don't. Moving forward from 1938 onward all non-German/non-aryans had to wear a badge domestically and in the conquered territories. Different colours signified what creed, race, religion, or "affliction" you had so it wasn't just the Jewish people who wore them.

Think about trying to "blend" in considering all the above. The Jewish population was largely already part of its own community now with additional economic and welfare restrictions in place. You couldn't stay in the Jewish community, so where do you live now? How do you get the money? Where do you get fake papers? How do you get your entire family to blend in? How do you prove your ancestry if the need arises? You're not even safe if you're married. The nazi's went from being the main political party to being an absolute, totalitarian dictatorship in less than a year. Think how long it takes to get a pothole fixed in the road by your council - the speed at which things progressed in the government was unprecedented, and the Jewish people, already heavily restricted, had little/no chance to react. The legislature and threat of violence made it nearly impossible without the risk of cruel and unusual punishment. From the early 1930s the nazi's were narrowing down their search area ever farther with each piece of legislation and each action. This all leads to the holocaust, "the final solution", and the horrible events that were to follow.

TL;DR - Enacted legislation to: restrict economic access, define what is a Jewish person, restrict access to state and social services, restrict contact with christian/aryans. Then conducted census using said legislation to identify Jewish people, proceeded to steadily unleash violence and sadism on a new scale.

I did a paper as an undergrad in a holocaust seminar class about what things were like in a given country prior to the Nazis. I'm of mainly Hungarian Jewish ancestry so I picked Hungary. One of the professors at my school was a Hungarian Jew who was a child during that time and I interviewed him.

He mentioned what saved him and his family is they were not very religious, so he had never gotten circumcised. They were hiding with a few other families when the Arrow Cross came. It was explained the families were just scared, that's why they were hiding. To see if people were Jewish or not, they had all the boys expose themselves to see if they were circumcised or not, because that was a strictly Jewish practice. Since my professor's family didn't do that even though they were Jewish, they were let go because it was assumed they were not Jewish.

A combination of a lot of things. I'd say the largest contributor early on was people identifying themselves as Jewish because they feared repercussions if the Nazis found out they were lying. Also, stubborn pride in their beliefs. Imagine in some bizarre world where Canadian Muslims invaded the United States and shipped american Baptists to concentration camps. You know as well as I do that a ton of them would not back back down from their beliefs, even if just pretending to save the lives of them and their families. Martyrdom under persecution is celebrated in Judaism and Christianity alike.

So those that did not escape the continent, turn themselves in, or martyr themselves, got taken in a number of ways. They used the national census (Nuremberg Laws required citizens to declare a race, and you guessed it, "Jews" was one of the races) as well as ledgers taken from synagogues with parishioner names, and many were sold out by their friends and neighbors who either feared repercussions or agreed with the extremism. Going back to another America comparison example, look how many people ran to report immigrants to authorities, many of who were here legally.

Finally, it wasn't just Jews. Jews just tended to display what Nazis deemed "undesirable" physical traits (i.e. not white) and happened to be living in the countries the Nazis invaded. Probably a large reason as to Hitler's disdain towards them similar to how some racist Americans hate Mexicans just for existing. It was mostly Jews, but Nazis rounded up anyone that had traits they deemed undesirable...Jews, Africans, Middle Easterners, East Asians, Gypsies, mentally and physically handicapped, etc. The Nazis worked with the Arab, African, Asian, Russian, and Mediterranean nations out of necessity, but still saw them as servile subhumans similar to the Jews (although Jews and Gypsies were considered the bottom rung of Hitler's racial ladder described in Mein Kampf). However these groups controlled parts of Europe, Africa, and Asia that Hitler needed access to, but there is no doubt he would have enslaved/murdered every last one of them if the Third Reich would have survived long enough to conquer all of Europe. Their goal was racial purity, and that wouldn't be possible with dark complected, or Asian people still left on Earth.

This is illustrated by thSoviet Union. A lot of people don't fully understand the full scope of the Soviet Union's influence on WWII. It's super complex geopolitics, and a lot of it is likely lost with time, but the short of it is that Soviet Russia came to a nonaggression pact with Germany in 1939 under the premise that they'd be awarded land in Poland, Romania and the Baltics if they basically protected the German East flank. However, Hitler began planning the Russian invasion probably long before the agreement, and that didn't stop him. Germany ended up invading Russia a few years later after Russia invaded Finland, lost a fuckton of military personnel to the numerically and technologically inferior Finnish people, and exposed their tactical weaknesses, and Germany used their weakened position to invade in order to kill off the Slavic people and replace them with Germans. Had Hitler waited a few years or continued his alliance with Stalin, most of the world would probably be speaking German and Russian. But anyways, that's how Hitler treated his "racially inferior" allies. Because to him, Slavic and Aryan, although both Caucasian, were two completely different races with Slavs being the inferior of the two.

On an unrelated aside, D-Day served a dual purpose that a lot of people don't know about. It wasn't just an invasion of German-occupied Europe and Africa for the sake of liberating those countries. Although Russia did a huge amount of damage to German forces during the German offensive (and took huge losses of over 20 million too), Stalin was an ambitious man. After the German retreat from Russia from 1942-1943, the allied powers were concerned that the Soviets would follow at the heals of the German retreat and claim abandoned nations as new additions to the Soviet empire. D-Day was as much about the race to Germany to prevent USSR occupation as it was about putting an end to the Nazi regime.