The "Final Solution to the Jewish Question" (in German, Die Endlösung der Judenfrage) was the euphemism used by Nazi officers to refer to Nazi plans to exterminate all Jews in Europe — what came to be known to the rest of the world as the Holocaust.
What was the "final solution" in the book Night by Elie Wiesel?
Throughout World War II, Jews were punished, boycotted, discriminated against, and murdered by Nazi soldiers, especially within Germany. But it wasn't until around 1941 that the total systematic annihilation of the Jewish peoples of Europe became a policy of Hitler's army. Later, this policy was "officially" known by the code name Operation Reinhard. Operation Reinhard began with the construction of three death camps in German-occupied Poland and ultimately ended in the murder of 6 million Jewish men, women, and children, as well as homosexuals, gypsies, Slavs, the handicapped, and anyone else who didn't fit in with Hitler's idea of a "pure" race of man.
You can find out more about Hitler's "final solution" in the historical background presented in CliffsNotes for The Diary of Anne Frank.