The Holocaust is a genocide of special and unprecedented type. The word genocide was coined by Raphael Lemkin, a refugee Polish-U.S. Jewish lawyer in 1942. While genocides are planned attempts to destroy ethnic or racial groups, the Holocaust was a radical attempt to annihilate every single member of the Jewish race.
The Holocaust was purely ideological. The Nazis imagined an international Jewish conspiracy to control the world (think "Protocols of the Learned Elders of Zion"). More on the fictitious Protocols another time. Other genocides have had practical motivations.
The Holocaust was not geographically limited. Since the Nazis fully intended to rule the world, directly or through allies, Jews around the world were to be hunted down and exterminated.
The Nazis were looking for all Jews. Persons with three or four Jewish grandparents were sentenced to death for the crime of having been born.
The Holocaust dehumanized the perpetrators and humiliated the victims.
In Nazi thinking all humanity was to be reorganized. The so-called Aryan race, they believed, was destined to dominate the world. All other races or social groups were to be either abolished or subordinated.
Finally, to accomplish this Nazis combined the ideology with modern technology and record keeping. Ultimately, this combination made it possible to exterminate 2/3 of European Jewry in less than a decade, together with millions of Slavic and other peoples.
Connected to this, in his 2006 argument Pope Benedict claimed that the "ultimate Nazi target" was the Christian church. If the tap root of the Christian faith, i.e. Judaism, could be torn up, the church might also be destroyed, he said.
This statement seems to ignore the nearly two millenia-long antisemitic hatred sponsored by many within the Christian Church who maintained that the Jews were responsible for rejecting and murdering their long-awaited Messiah. Since the Jews asked that Jesus blood be upon them and their children (Matt.27:24-25), let it be so. Their old covenant has been superseded by the New Covenant in Christ.
Related to this issue is the Roman Catholic Church's more recent affirmation of the permanent integrity of God's ancient covenant with the Jewish people. Many scholars, including some Lutheran, have insisted that this is so and are calling for Christians to reevaluate their relationship to the Jewish people.
These issues must be explored in greater detail another time.