The term "reich" means "kingdom" or "empire," so Third Reich means Third Kingdom. The Third Reich followed the first two reichs in German history.
The Third Reich was established by Adolf Hitler in 1933. This followed the First Reich, which ended in 1806, and the Second Reich, which came about in 1871 and lasted until 1918. The Third Reich crumbled in 1945 at the end of World War II.
Before Hitler established the Third Reich, the Weimar Republic, Germany's democratic government, had to be destroyed. The end of the Weimar Republic came about because of political deadlock and economic depression. In 1933, a suspicious fire broke out in the German Parliament that caused the German citizens to overthrow the current government.
Once the government had crumbled, Hitler seized the opportunity to reach out the the German people who desperately wanted a new, more effective form of government. It wasn't long before Hitler started calling his government the Third Reich and everything from law to education were under Nazi control. Through carefully orchestrated public speeches and propaganda, Germany was soon under Hitler's spell, and he was charismatic enough to have everyone from wealthy, influential people to the homeless rallying behind him.