Reich is the German word for "realm" or "empire". It is the word traditionally used for sovereign entities, including Germany. For instance, any country with a King or Queen as head of state, such as Britain, is a Königreich (kingdom).
The term Reich was part of the official names of Germany over centuries. The German name for "Holy Roman Empire of the German Nation" (mid 10th century - 1806) is Heiliges Römisches Reich Deutscher Nation. Later, Deutsches Reich was the official name of Germany from 1871 to 1945, although its three very different political systems are commonly referred to as German Empire (1871-1918), Weimar Republic (1919-1933), and finally Nazi Germany (1933-1945).
The Nazis, eager to present their rule as a continuation of a Germanic past, used the term Das Dritte Reich ('The Third Reich'), counting the Holy Roman Empire as the first and the 1871 Empire as the second. They also used the political slogan "Ein Volk, ein Reich, ein Führer" ("One people, one country, one leader"). A number of words used by the Nazis which earlier were neutral have later taken on a negative connotation; the word "Reich" is one of them. Some right-wing extremists in Germany use the term today to describe their country.