The term "Final Solution" refers to their deliberate policy of exterminating as many of Europe's Jews as possible.
The expression was used to fit in with Nazi propaganda that they were solving an alleged problem - that is Jews and other races were racially undesirable and were weakening the gene pool and deliberately preventing the Germanic/Aryan race from fulfilling its true potential and destiny. Also because if they said that they were going to gas several million Jewish people that the policy might have been unpalatable to many people.
The stages in the Final Solution were:
1933 Hitler appointed Chancellor and via the Enabling Act he took dictatorial powers.
1935: Nuremberg laws - these defined who was and wasn't Jewish, restricted certain rights to Jewish people, such as making them a Star of David, banned Aryans and Jews from getting married.
1938: Kristallnacht - the night of broken glass, following the assassination of a German embassy official by a French-Polish Jew upset at his family's deportation from Germany to Poland an antisemitic pogrom was carried out where synagogues were burnt down, Jewish shops and businesses attacked and many Jewish people attacked leading to many deaths. After Kristallnacht the Jewish community was made to then pay for the damage.
1939: WWII kicks off with the Nazi invasion of Poland. Following up behind the invasion were specially appointed groups of SS thugs called Einsatzgruppen. Their job was to round up groups of Jewish people and execute them. The invasion of Poland and then the USSR after the invasion of the Soviet Union brought millions of Jewish people into the Nazi Empire. Those who the Einsatzgruppen did not murder were placed in ghettos and they were forced to work in Nazi related industries - making uniforms, crushing rocks for road construction etc. However the large number of Jewish people meant that all other options that were debated such as letting them die off gradually, deporting them to Palestine, Madagascar etc were considered impractical as Germany was at war.
20 January 1942 - The Wannsee Conference - under the chairmanship of Reinhardt Heydrich a group of Senior Nazis endorsed what became known as the Final Solution or the Holocaust and meant the genocidal death of the Jews in Nazi captivity. It was estimated at this conference that there were 11 million Jews in Nazi occupied Europe.
Subsequently a number of extermination camps, mostly in Poland, were established with the purpose of gassing as many Jews as possible via the use of Zyklon-B gas pellets. These were dropped into sealed rooms designed to look like shower rooms and were activated by human body heat. The bodies, after they were processed, that is gold fillings were removed as well as prosthetic limbs etc were then mostly cremated.