Saw this question on another site and got curious.
Your question is essentially the crux of the first major historiographic debate in Holocaust studies, the question of intentionalism vs. functionalism. The key disagreement between the two schools of thought was how early the Nazis had come up with the plan to systematically exterminate the Jews of Europe. Intentionalists argued that Hitler had planned to carry out the Final Solution well in advance (in some interpretations, even before he came to power), while functionalists argued that there was no such master plan and the Final Solution developed through a process of gradual radicalization. This debate isn't necessarily settled, but the consensus of most historians has converged on a moderate functionalist interpretation of the Holocaust.
I'm not sure if you're interested more in the historiographic debate between the intentionalist and functionalist schools or if you're just interested in the current historical consensus, but answering the former kind of helps to explain the latter. The seminal work of Holocaust history was Raul Hilberg's The Destruction of the European Jews, which was published in 1961. It wasn't the first book about the Holocaust, but it was the first systematic attempt to document it and it kicked off the debate between functionalism and intentionalism. Hilberg's work established the functionalist school: he argued that the Holocaust was the result of a gradual process of radicalization, beginning with the antisemitic laws adopted shortly after the Nazis came to power, to economic and social discrimination, to increasingly radical rhetoric which dehumanized the Jews, to ghettoization, to actual physical destruction. The progression from discrimination to extermination laid out in Hilberg's work actually closely mirrored the ten stages of genocide paradigm, which was created 25 years later. Hilberg's book was the seminal work that really established Holocaust studies as a separate field of history and his arguments began the debate between functionalist and intentionalist historians.
Some historians subsequently took Hilberg's functionalist arguments to the extreme, including Martin Broszat and Hans Mommsen, who claimed that the Nazi leadership had little to do with the decision-making during the Holocaust, with the initiative coming almost entirely from below. This argument didn't get much credence from most historians, but it does illustrate an important concept that's now a key part of the mainstream interpretation of the Holocaust: the idea of "working toward the Führer". This concept, which was most notably argued by Christopher Browning, posits that the driving force behind the Holocaust was the massive, unstable bureaucracy of Nazi Germany, which featured a number of ambitious actors who wanted to curry favor with Hitler. In the absence of explicit instructions from Hitler, they tended to do this by taking his words and rhetoric to their logical extreme. When this approach was applied to the "Jewish question", it inevitably led to a progressive radicalization that culminated in the Final Solution.
The intentionalist argument, by contrast, posited that Hitler had a master plan to carry out the Final Solution and that he directed it, rather than his subordinates. Historians like Lucy Dawidowicz and Andreas Hillgruber argued that Hitler had such a master plan even before he came to power and even before he wrote Mein Kampf. More moderate intentionlists argued that Hitler only developed this master plan after coming to power in the 1930s. The arguments of the intentionalist school are hampered by the lack of documentary evidence that Hitler directly orchestrated the Holocaust. It's possible that such evidence exists and was lost or deliberately destroyed, but the lack of documentation for the intentionalists' central claim kind of undercut their arguments. In German historiography, the functionalist-intentionalist debate got further complicated in the 1980s by a larger historiographic conflict over the uniqueness of the Holocaust and the question of Germany's collective guilt; this conflict is known as the Historikerstreit (historians' conflict), and I'm not going to get into it here because it's complicated and also not really what your question is about.
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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
While more can be said about this, you may be interested in this previous answer about this question from u/commiespaceinvader