The Third Reich (German: "Empire") was, as is commonly known, the Germanic state promoted by Hitler and the Nazi Party from 1933-1945.
The First Reich was the Holy Roman Reich (usually translated as "Holy Roman Empire"), a more-or-less Germanic federation lasting from 800-1806. *
The Second Reich was the so-called German Empire formed by Otto von Bismarck in 1871. World War I (1914-1918) brought about the rapid demise of this Reich.
Editorial: Maybe it's just me, but I think the Second Reich suffers from the middle-child syndrome. Lots of people have heard of the Holy Roman Empire, and almost everybody knows of the Third Reich, but the Second Reich - the forgotten middle child - kind of gets brushed under the rug. Not only did it last far longer than the Third Reich (47 years vs. 12 years), but it too had it's own World War (not that that's a good thing). Ohh well. Did I mention that I'm a middle child...
* Holy Roman Empire
German: Heiliges Roemisches Reich Deutscher Nation (thanks WoOS)
Latin: Sacrum Romanum Imperium