It means 'empire', specifically the German State. The First Reich was the Holy Roman Empire, the Second Reich was the empire of the Kaisers and the Third Reich was Nazi Germany.
Another interesting question is why do we say Third Reich and not Third Empire (or Dritte Reich)? There's at least one other example of mixing up English and German as we also say Bayern Munich (why is not Bavarian Munich?). Does no-one else find these constructions strange?
Yes. I have always wondered why the media people don't make up their minds and say "Bavaria Munich" or "Bayern Munchen". I think that "Reich" has a more specific meaning than just "Empire" because it denotes the Nazi superstate which did not have an Emperor. The one I dislike is the use of the word "Fuhrer" which is left untranslated as if to imply that it has a special meaning. In reality, the word "Fuhrer" simply means "leader" (just as "duce" and "taoiseach" also mean leader). If I ever made a film about the Nazi period, I would have the script using "The Leader" and "My Leader" rather than "Fuhrer".
A similar situation relates to AC Milan. AC stands for 'Associazione circlio' - football club, but Milan is English for Milano.
This was deliberate, as it was originally Milan FC, as set up by a British man. Mussolini insisted it used Italian, so was changed to AC, but somehow left the Milan bit alone.