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Hi! As this question pertains to basic, underlying facts of the Holocaust, I hope you can appreciate that it can be a fraught subject to deal with. While we want people to get the answers they are looking for, we also remain very conscious that threads of this nature can attract the very wrong kind of response. As such, this message is not intended to provide you with all of the answers, but simply to address some of the basic facts, as well as Holocaust Denial, and provide a short list of introductory reading. There is always more than can be said, but we hope this is a good starting point for you.
##What Was the Holocaust?
The Holocaust refers the genocidal deaths of 5-6 million European Jews carried out systematically by Nazi Germany as part of targeted policies of persecution and extermination during World War II. Some historians will also include the deaths of the Roma, Communists, Mentally Disabled, and other groups targeted by Nazi policies, which brings the total number of deaths to 11-17 million. Debates about whether or not the Holocaust includes these deaths or not is a matter of definitions, but in no way a reflection on dispute that they occurred.
##But This Guy Says Otherwise!
Unfortunately, there is a small, but at times vocal, minority of persons who fall into the category of Holocaust Denial, attempting to minimize the deaths by orders of magnitude, impugn well proven facts, or even claim that the Holocaust is entirely a fabrication and never happened. Although they often self-style themselves as "Revisionists", they are not correctly described by the title. While revisionism is not inherently a dirty word, actual revision, to quote Michael Shermer, "entails refinement of detailed knowledge about events, rarely complete denial of the events themselves, and certainly not denial of the cumulation of events known as the Holocaust."
It is absolutely true that were you to read a book written in 1950 or so, you would find information which any decent scholar today might reject, and that is the result of good revisionism. But these changes, which even can be quite large, such as the reassessment of deaths at Auschwitz from ~4 million to ~1 million, are done within the bounds of respected, academic study, and reflect decades of work that builds upon the work of previous scholars, and certainly does not willfully disregard documented evidence and recollections. There are still plenty of questions within Holocaust Studies that are debated by scholars, and there may still be more out there for us to discover, and revise, but when it comes to the basic facts, there is simply no valid argument against them.
##So What Are the Basics?
Beginning with their rise to power in the 1930s, the Nazi Party, headed by Adolf Hitler, implemented a series of anti-Jewish policies within Germany, marginalizing Jews within society more and more, stripping them of their wealth, livelihoods, and their dignity. With the invasion of Poland in 1939, the number of Jews under Nazi control reached into the millions, and this number would again increase with the invasion of the Soviet Union in 1941. Shortly after the invasion of Poland, the Germans started to confine the Jewish population into squalid ghettos. After several plans on how to rid Europe of the Jews that all proved unfeasible, by the time of the invasion of the Soviet Union in 1941, ideological (Antisemitism) and pragmatic (Resources) considerations lead to mass-killings becoming the only viable option in the minds of the Nazi leadership. First only practiced in the USSR, it was influential groups such as the SS and the administration of the General Government that pushed to expand the killing operations to all of Europe and sometime at the end of 1941 met with Hitler’s approval.
The early killings were carried out foremost by the Einsatzgruppen, paramilitary groups organized under the aegis of the SS and tasked with carrying out the mass killings of Jews, Communists, and other 'undesirable elements' in the wake of the German military's advance. In what is often termed the 'Holocaust by Bullet', the Einsatzgruppen, with the assistance of the Wehrmacht, the SD, the Security Police, as well as local collaborators, would kill roughly two million persons, over half of them Jews. Most killings were carried out with mass shootings, but other methods such as gas vans - intended to spare the killers the trauma of shooting so many persons day after day - were utilized too.
By early 1942, the "Final Solution" to the so-called "Jewish Question" was essentially finalized at the Wannsee Conference under the direction of Reinhard Heydrich, where the plan to eliminate the Jewish population of Europe using a series of extermination camps set up in occupied Poland was presented and met with approval.
Construction of extermination camps had already begun the previous fall, and mass extermination, mostly as part of 'Operation Reinhard', had began operation by spring of 1942. Roughly 2 million persons, nearly all Jewish men, women, and children, were immediately gassed upon arrival at Bełżec, Sobibór, and Treblinka over the next two years, when these "Reinhard" camps were closed and razed. More victims would meet their fate in additional extermination camps such as Chełmno, but most infamously at Auschwitz-Birkenau, where slightly over 1 million persons, mostly Jews, died. Under the plan set forth at Wannsee, exterminations were hardly limited to the Jews of Poland, but rather Jews from all over Europe were rounded up and sent east by rail like cattle to the slaughter. Although the victims of the Reinhard Camps were originally buried, they would later be exhumed and cremated, and cremation of the victims was normal procedure at later camps such as Auschwitz.
##The Camps
There were two main types of camps run by Nazi Germany, which is sometimes a source of confusion. Concentration Camps were well known means of extrajudicial control implemented by the Nazis shortly after taking power, beginning with the construction of Dachau in 1933. Political opponents of all type, not just Jews, could find themselves imprisoned in these camps during the pre-war years, and while conditions were often brutal and squalid, and numerous deaths did occur from mistreatment, they were not usually a death sentence and the population fluctuated greatly. Although Concentration Camps were later made part of the 'Final Solution', their purpose was not as immediate extermination centers. Some were 'way stations', and others were work camps, where Germany intended to eke out every last bit of productivity from them through what was known as "extermination through labor". Jews and other undesirable elements, if deemed healthy enough to work, could find themselves spared for a time and "allowed" to toil away like slaves until their usefulness was at an end.
Although some Concentration Camps, such as Mauthausen, did include small gas chambers, mass gassing was not the primary purpose of the camp. Many camps, becoming extremely overcrowded, nevertheless resulted in the deaths of tens of thousands of inhabitants due to the outbreak of diseases such as typhus, or starvation, all of which the camp administrations did little to prevent. Bergen-Belsen, which was not a work camp but rather served as something of a way station for prisoners of the camp systems being moved about, is perhaps one of the most infamous of camps on this count, saw some 50,000 deaths caused by the conditions. Often located in the Reich, camps liberated by the Western forces were exclusively Concentration Camps, and many survivor testimonies come from these camps.
The Concentration Camps are contrasted with the Extermination Camps, which were purpose built for mass killing, with large gas chambers and later on, crematoria, but little or no facilities for inmates. Often they were disguised with false facades to lull the new arrivals into a false sense of security, even though rumors were of course rife for the fate that awaited the deportees. Almost all arrivals were killed upon arrival at these camps, and in many cases the number of survivors numbered in the single digits, such as at Bełżec, where only seven Jews, forced to assist in operation of the camp, were alive after the war.
Several camps, however, were 'Hybrids' of both types, the most famous being Auschwitz, which was a vast complex of subcamps. The infamous 'selection' of prisoners, conducted by SS doctors upon arrival, meant life or death, with those deemed unsuited for labor immediately gassed and the more healthy and robust given at least temporary reprieve. The death count at Auschwitz numbered around 1 million, but it is also the source of many survivor testimonies.
##How Do We Know?
Running through the evidence piece by piece would take more space than we have here, but suffice to say, there is a lot of evidence, and not just the (mountains of) survivor testimony. We have testimonies and writings from many who participated, as well German documentation of the programs. This site catalogs some of the evidence we have for mass extermination as it relates to Auschwitz. I'll end this with a short list of excellent works that should help to introduce you to various aspects of Holocaust study.
##Further Reading
The main reasons were manpower, logistics and especially the psychological effect a murder has on a person. Also, while the leadership such as Goebbels and Himmler could've been psychopathic, it was the soldiers pulling the trigger.
Manpower is the reason why they created Vlasov's army.
Logistics-wise, they didn't want to waste bullets on civilians.
Logic-wise, the mass murders and genocides did occur, but only in the beginning of the war, before Soviets started pushing back. At that point any Drang Nach Osten and the Generalplan Ost plan was deemed illogical, as it was highly likely the Third Reich wouldn't be there to reap the benefits of unpopulated lands.
Now for the psychological effect: As civilizations develop, we are less and less exposed to violence, making us more and more fragile when we see one. Imagine seeing bunnies. They make for cute pets, but while it's a pet for the majority of Western Europeans and especially Americans (this is an interesting article https://www.theatlantic.com/business/archive/2014/08/are-rabbits-pets-or-meat/378757/), in Eastern Europe it still is somewhat popular meat in Eastern Europe, although an expensive one. They are bred to be killed and eaten in these parts of lands, not to be kept as pets. This is due to the fact that back in Socialism we were able to own mainly chicken, rabbits, sheep and other animals, and having a rabbit in agricultural countries that you could feed with grass that grows in your garden proved to be ridiculously effective (unlike chicken).
Now it may seem like a silly analogy, but it shows us that even the cute animals are killed for meat here in EE, while in the more westernized part of world they became too cute to be killed and have instead become mainly gorgeous pets with a soft fur. Meanwhile in other parts, much less Westernized, dogs are eaten as well, beaten and skinned alive so the meat would arguably taste better. Something unspeakable for Europeans. Monkey brains are eaten. The list goes on. This is however largely deemed as a cultural phenomenon, but mark my words, as those countries become more developed, they'll shy away from these things and they will disappear. It's not because they've become Europeanized, but that they've decided they make for better pets and that eating monkeys' brain is just pointless. As they'll become more developed, their emotional quocient and empathy will rise. Monkeys will be more sympatized with, dogs will be too cute and too loyal to be eaten just like bunnies became in Western Europe and horses in Eastern Europe (though Polish meat producers still consider them as a good market).
Now back to the war: The rougher your life is, the less likely you're going to break at stressful events.
The more you're exposed to death and violence, your senses become duller and duller. Germany was highly industrialized even in WW1, its population urbanized. Not many saw the brutal farm life of innocent animals being slaughtered, blood spilling from their slashed necks, most were behind desks at schools or working behind assembly lines, in the press, shoe industry, etc. Suddenly you, a journalist's son, fight in a war, you kill soldiers and you argue they would've killed you if you hadn't pulled that trigger. But the numbers keep rising and you question your sanity. Then you're ordered to shoot civilians and see as their lifeless bodies fall into a mass grave. That was a person with a family, a name, favorite meals and activities, friends, that person could've had children had you not pulled the trigger. Was his different culture the right reason to kill him? A sane person says no.
Psychopaths do not feel empathy, sociopaths are more empathic but in general these two types of people would be the only ones who wouldn't feel that much of a remorse. They are just broken from the start and do not see anything bad about themselves doing this. Psychopaths are around 1% of the population. That means that the militias doing these hyenous acts would sooner or later have PTSDs and other irreversible and debilitating psychological disorders. It means that particular person is not fit to fight in a war anymore. In terms of war it's basically equivalent to a POW, KIA, MIA or seriously wounded person. No one would want that, especially in the middle of the greatest war humanity has ever seen.
Like I previously said it came to that, systematic mass killings of Slavs occurred, but most Slavic civilians that were killed (over 20 million in the RSFR alone) were starved to death, died of diseases or to bombardment. You don't see much of that death as a soldier. They besieged Leningrad where 1 million civilians died in the streets of that city. Not many people got PTSDs from that because they weren't there to see it, they couldn't consider themselves responsible, only those very empathetic and very emotional did - those must've been very, very rare cases. But nonetheless these bombardments and starvations of cities was a part of Generalplan Ost. Much more effective than rounding up villagers and city dwellers and making some poor Munich-dweller pull the trigger.
On the other hand, imagine Soviet soldiers. Recapturing your father's farmhouse only to see it burned, family nowhere to be seen. You are filled with unsatiable rage, you crave for vengeance and fear is no longer a factor. You want them to pay, no matter who comes across your sights, they all fought for this to happen. You want them to pay. Eastern Europeans on top of that were not urbanized nor industrialized with the exception of Bohemia. That meant most people were living rough farmer lives before Germans paid their visit, seeing death as children, more frequently both in terms of human losses (especially in the USSR, during revolution and shortly after during Holodomor in Russia and then in Ukraine) and in terms of animals and pets.
The psychological effect of killing soldiers or civilians was not as debilitating for the Soviets (and other Slavs fighting in that war), as they chances are, they were already either broken or had a different psyche and life philosophy that saved them from breaking, but marked them for life instead. It was because of their past life and because of what they saw at home that they became dulled in their senses. One should not underestimate the effect vengeance has on people. Many people have vengeance as their only sense of purpose and the Soviet propaganda was powerful, channeling that energy to fight off the Germans and eventually winning. The only goal was to win, and payback was the ultimate cherry on the cake. While rapes were exaggerated, I am fairly certain they occurred more by the Soviets than the Germans, especially because of things mentioned above.