The "Final Solution" was the Nazi term for the extermination of the Jews. The term "Final Solution" is short for the phrase "final solution to the Jewish problem."
To the Nazis, the presence of Jews in German territory was a problem. The Nazis believed that Jews were an inferior "race" whose presence would lead to the decay and eventual destruction of German society. Because of this, they wanted to get rid of the Jews.
There were a number of ideas proposed and methods tried for getting rid of Jews. These included ideas such as forcing Jews to emigrate to Madagascar and methods such as simply shooting Jews, as was done by the Einstazgruppen in Eastern Europe. However, none of these was satisfactory, for various reasons. Eventually, the Nazis hit upon the system of extermination camps. This, they believed, was the final solution to the Jewish problem.