The term or concept of a "Third Reich" puzzles many people and we're often asked about it, so here's a quick overview. We'll start with "Reich", move on to the counting up to "the third" and finish with the "Third Reich" as a Christian concept.
...is just an old German noun for a dominion/realm without further specification. In connection with another noun we get "Kaiserreich" for an empire (Kaiser is an emperor or "Caesar"), Königreich for a kingdom (König is king) or Frankreich for France.
When referring to Nazi Germany as the Third Reich, the first is the so-called Holy Roman Empire (Heiliges Römisches Reich) which was told to have been established in 800 and ceased to exist in 1806 (hence Hitler's vision of another one-thousand-years Reich). Just as an example, these were its borders in 1648:
The second Reich in this counting begins with the creation of modern Germany in 1871 under leadership of the Prussian kingdom, whose king became emperor of the new German Empire or Deutsches Reich:
However, this new empire didn't refer to itself as the "second". Since there hadn't been yet a "third empire", it was simply referred to as the New Empire. Accordingly, its "Holy Roman" predecessor wasn't seen yet as "the first", but simply as the Old Empire. Only the end of this New Reich in 1918/1919 made it possible to think of the future one as the third. In other words, referring to Nazi Germany as the Third Reich is exactly what turned the Holy Roman Empire into the first and the modern German Empire into the second Reich, nicely placing low-class Hitler as a worthy heir to the royal dynasties of the previous empires in this new trinity.
Speaking of trinity, the "Third Reich" is not just a product of modern historical retrospect in the 20th century, but a much earlier term of Christian salvation history, along the lines of the Old Testament as the first, the New Testament as the second and current era, and the savior's or divine Return as the third Reich, the future heavenly kingdom. In this context, the application of this already existing religious term to Nazi Germany can be seen in two ways: as secularization of Christian salvation history, in which Hitler replaces Jesus like Socialism had replaced Christianity before; but simultaneously also sacralization of modern politics, bestowing it with some holiness and placing Hitler under divine providence as Germany's savior (which could make one wonder if Christianity might have been the first personality cult, but that's a whole other realm).