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
I wouldn't say that the original answer is an example of a Holocaust denial, although one of the followups definitely is. Let's address them in that order, although I would rather start from the question that introduces some misconceptions from the start. I'll allow myself to use a former responses to the questions from other subreddits.
In general, Hitler and most if not all of the Nazi decision-makers did not consider Jews 'evil'. The National Socialist ideology was based on the ultra-nationalist principles that, due to their volkist roots took much more racist shape than many other nationalist policies. Thus, the Jews, much like the Slavs, were considered 'unfit' to be citizens of the new German state on account of their ethnic origins. Hitler's main political argument behind the anti-Semitic policies was the fact that due to his experiences, he tended to associate German and Austrian Jews with either communism or social democracy (both attracting educated youth and middle class respectively, and relatively attractive to Jews due to their intrinsic inclusiveness), political orientations irreconcilable with National Socialism that was staunchly anti-communist and anti-democratic. Furthermore, Hitler noticed that Jews, despite living in various countries, maintained very high level of national and cultural identity. This, compounded with the development of Zionist movement led him to believe that Jews do not want to integrate, prefer cultural separation and vie to form a separate Jewish state. This meant, that in Hitler's opinion, Jews would not make 'good Germans' and thus their presence in Germany might be detrimental to the 'national unity' what, of course, became even more important with the gradual development of the Germanic supremacist theories. These observations also laid at the base of the general animosity towards the Jews and the accusations of sabotaging the war effort after the First World War (a scapegoating that bears uncanny resemblance to the medieval anti-Semitic myths about poisoning the wells or spreading plague). The latter, although not that popular per se, was quite prominent among Pan-Germanists and nationalists. These notions date back several decades, being raised by people like e.g. Eugen Dühring, German philosopher and sociologist who called for the eradication of Judaism through social development, laying groundwork for volkist policies used extensively by NSDAP. He openly advocated annihilation of Jews as an ethnic group in an international, coordinated manner, pointing out that mere deportation won't help much, because Jews will still be able to influence the affairs of the other countries. In his 1907 article Sociale Rettung durch wirkliches Recht statt Raubpolitik und Knechtsjuristerei (Social rescue through the real law instead of cleptocracy and venal judicial power) that the 'Jewish socialism is responsible for the stoking the class struggle and will be met with retaliation in the form of the racial war'. So, in a way Hitler did not create the anti-Semitic rhetoric to ascend to power, but rather used the pre-existing prejudices to attract anti-Semites and ultranationalists to his cause. And as we can see, Jews were generally targeted as as the entire ethno-religious group, so even if National Socialists were giving credit to the conspiracy theories about the 'International Jewry controlling the world' (that were surfacing since the late 19th century), they did not limit the persecution to the Jews in influential positions. Anti-Semitic laws generaly targeted all the Jews equally.
Now for the second answer that is a direct Holocaust denial, resulting from ill will or ignorance on the part of its author (although given the knowledge of the treatment of Jews in Germany I strongly suspect the former). And as strange as it sounds, such misconceptions are still quite common, what is pretty interesting, given that the Holocaust is a topic that has seemingly been covered very thoroughly.
It is worth to be noting The information provided by Mr. Yeteln is generally correct, but it omits the simple fact that Holocaust largely did not happen in Germany and thus rendering the whole argument invalid. It is true that the 'Jewish question' was far smaller than Nazis made it to be. Germany in late 1930s (after the annexation of Austria and Sudetenland) was a home almost 70 millions citizens, some 550.000 of whom were Jews, what amounts to 0.8%, almost a negligible number. And with the progressively more and more hostile politics, many of these indeed emigrated from Germany even before proclamation the anti-Semitic regulations (e.g. so-called Nuremberg Laws of 1935) and intense physical persecution of the Crystal Night. It should be noted that Hitler himself was involved in brokering deal with Palestinian organization to facilitate resettlement that eventually was realized as 'Transfer agreement' (heb. heskem haavara) allowing some 50.000-60.000 German Jews to emigrate to British Mandate of Palestine circumventing British immigration restriction. It is estimated that shortly before the war, in Germany resided roughly 200.000 Jews. Thus, it is clear that Hitler and Nazis in general were initially removal of Jews from the territory of Germany rather than their extermination and their plan largely succeeded. But the war complicated things to say the least.
Given that the number of Jews who perished in the Holocaust is usually given as 5.5-6 millions, one might wonder where this discrepancy comes from. The truth is that vast majority of the Holocaust victims were Jewish inhabitants of the Central and Eastern Europe that created very large communities there, at least in comparison with the Western part of the continent. For example, total number of Jews in Poland was almost 3.5 million or 13% of the total population (making it also the biggest Jewish diaspora in a single country). To drive the point home, in mid-1939 population of Jewish Poles in Warsaw alone was one and a half times higher than the entire population of the German Jews in the Third Reich. And this was the largest part of the 'Jewish Question'. By 1939 Nazis basically coerced to leave, sequestered or outright imprisoned most of the Jewish inhabitants of Germany, but having conquered large part of Poland in 1939 and capturing its remaining part annexed by USSR in 1941, Germans paradoxically increased the number of Jewish citizens within their territory by 1750% (17.5 times) in comparison with 1938. It should be noted that although Poland was an undisputed centre of the European Jewish population, Jewish minorities were also quite large in e.g. Hungary where out of 9.2 million inhabitants 730.000 (8%) were Jews, or Lithuania, where 155.000 Jews formed 6.5% of the entire population numbering 2.4 millions. In other words, vast majority of the Holocaust victims were not Germans, but people living in the territories occupied by and, to lesser extent, also allied with Germany (about 3 millions were Polish nationals, another million were citizens of the Soviet Union, roughly 500.000 lived in Hungary and 280.000 in Romania). Thus, the staggering 97.5% of the Holocaust victims were not Germans and thus the situation of the German Jews are by no means representative to the Holocaust as such, especially in the pre-war period.
Initially, German plans regarding Jewish inhabitants of the Third Reich, including the conquered territories focused on the possibility of a forcible resettlement to some remote location or even creation of the Jewish enclaves (this was also extended to e.g. Poles, with Brazil being selected as a possible destination). This idea was predating NSDAP by several decades, having been first proposed in 1885 by Paul Anton Bötticher in his article The most urgent duties of German politics and were picked up both by Zionist groups that saw it as a opportunity to create 'land without nation for the nation without land' and the anti-Semites who considered it an opportunity to remove Jews from the territory of Germany. In 1933, German authorities made an agreement with German Zionist organizations and Palestine authorities, allowing mass emigration of Jews to Palestine, circumventing the British immigration laws by presenting the idea (later known as Haavara Agreement) as an economic investment plan. This resulted in emigration of almost 10% of the German Jews to Palestine between 1933 and 1939, when the project was cancelled. One of the most common destination was considered to be some region of Eastern Siberia, as well as Poland, Palestine and Madagascar. The latter was considered after the conquest of France in 1940 (it also has a long history of a potential destination for the Jewish Zionists, considered by France, Great Britain and Poland that wanted to buy said colony from France for economic purposes), although the plans were scrapped after the failure to defeat Great Britain what with the naval superiority of Royal Navy made the idea unfeasible. Mass relocation to Siberia also became hardly accessible after the initially fast invasion of USSR came to a halt in late 1941 and it became obvious that the USSR won't 'fall apart after one kick in the door' to quote Hitler himself. This caused Nazis to simply consider the simplest if the most inhumane possibility, that is the physical elimination of the Jewish citizens.