"Reich" literally "rule" or "realm" is usually translated as "Empire" for the First and Second.
The "First Reich" is the Holy Roman Empire, from the crowning of Charlemagne on Christmas Day 800 until Emperor Francis II abdicated on August 6 1806.
The "Second Reich" is the German Empire, from the German unification on January 18 1871 until Wilhelm II abdicated on either November 9 (announced) or 28 (formally issued) 1918.
Sources:
Germany and the Holy Roman Empire by Joachim Bailey
The Long Nineteenth Century: A History of Germany, 1780–1918 by David Blackbourn
The Holy Roman Empire (of the German Nation) was the First, the German Empire of 1871-1918 the Second.
Ordering empires in steps of three is nothing new, in fact it goes back christian interpretations of scripture. Daniel 2, 37ff. says:
You, O king, the king of kings [...] you are the head of gold. 39 Another kingdom inferior to you shall arise after you, and yet a third kingdom of bronze, which shall rule over all the earth. 40 And there shall be a fourth kingdom, strong as iron, because iron breaks to pieces and shatters all things. And like iron that crushes, it shall break and crush all these. 41 And [...] it shall be a divided kingdom, but some of the firmness of iron shall be in it, just as you saw iron mixed with the soft clay. 42 And [...] the kingdom shall be partly strong and partly brittle. [...] 44 And in the days of those kings the God of heaven will set up a kingdom that shall never be destroyed, nor shall the kingdom be left to another people. It shall break in pieces all these kingdoms and bring them to an end, and it shall stand forever,
Daniel is here interpreting the dream of Nebuchadnezzar, King of Babylon. Now what Hieronymus does is directly interpreting these kingdoms, or 'Reiche' (the Latin text has regnum, which can be translated as Reich in German), with concrete historical Empires: The first is obviously the Babylonian Empire, followed by the Median or Persian Empire (Et post te consurget regnum aliud, minus te, Medorum videlicet atque Persarum), the Empire of Alexander and his diadochii (Et regnum tertium aliud [...] Alexandrum significat et regnum Macedonum successorumque Alexandri), with the Roman Empire being the fourth and final Empire, the Empire of iron (Regnum autem quartum, quod perspicue pertinet ad Romanos, ferreum est, quod comminuit et domat omnia); pointing out the parallels between these historical Empires and those mentioned in the Bible. So he's equating the divine ordering of the world with the historical 'order' of empires - the consequence being that the fourth Empire will end at judgement day and the coming of the Empire of God. This had several consequences, importantly the concept of the translatio imperii, the theory that the Roman Empire continued to exist even after the fall of Rome and later Constantinople in the form of the Holy Roman Empire. Since the End of the Roman Empire would have meant the end of times, obviously it would continue to exist in some form or other.
Now where does the importance of the Third Reich come from here? The Calabrian abbot Joachim of Fiore ordered history in steps of three, again based on biblical exegesis, now factoring in the New Testament. He interprets the biblical history as a blueprint for the history of salvation, dividing it in three parts modeled on the Holy Trinity: the Empire of the Father, the Empire of the Son, and the Empire of the Holy Spirit. The Empire of the Father was the world of the Old Testament, the empire of the son was the New Testament's world, and the third (which, after his prediction would begin in 1260) was to be the empire of the Holy Spirit, the New Jerusalem of the Book of Revelations. This he also called the 'Third Reich' - thus the Third Reich stood at the endpoint of salvation, the culmination of history in salvation. The Empire of Christ, which was the Fourth Empire of the earlier interpretation, would not be the end, but instead the Third Empire of the Holy Spirit. Each Empire would have a leading figure, and while Christ was that of the Second, the Third would have someone he labeled as 'dux' - which is 'Führer' in German.
The aftershocks of such teleological interpretations of history can be felt until Hegel and Marx, but suffice it to say that it was an influential view, and a dominant view of history in general until the Enlightenment and medieval salvation history turned into history made by Humans. Early socialists picked up the term as well.
So, how do we get from de Fiore's Third Reich of salvation, universal happiness and bliss to Hitler's Third Reich of destruction, genocide and war?
The term 'Drittes Reich/Third Reich' did not gain much traction in Germany until the beginning of the 20th century, but around that time it began to be used in nationalist circles. There's a similarity at the core of National Socialist ideology and christian, chiliastic beliefs. The National Socialist believed that their movement, and their revolution, would bring about a new and superior society, a salvation, centered on the Volk, the people and their community.
Dietrich Eckart was a largely unsuccesful writer and dramatist, but the driving force behind the early history of the NSDAP. Eckart demanded a revolution against the 'wage and interest slavery' (Zinsknechtschaft), wanted to destroy the anti-christian Soviet Union and 'drive the Jews into the sea'. He was an important early influence on Hitler, in his anti-semitism, anti-bolshevism, anti-republicanism and his revolutionary gnostic ideology; and Hitler himself named him one of his most important teachers. When the end of the German Empire came, and a socialist revolution seemed a possible threat, nationalistic ideologues began to posit a nationalistic empire as counterpoint to the threat of socialism. This was to be the 'Third Reich', a term Eckart probably picked up while translating a play of Henrik Ibsen. Eckart wrote, in 1919:
There is on Earth no other people that would be as capable of erecting the third Reich as the German people.
In this he unites his revolutionary political ideals with christian milleniarian expecations of salvation - the third Reich would be a nationalistic one. Eckart was deeply christian, and saw the German people as an especially christian one, capable as no other of opposing the socialist anti-christ. With this the term was part of the nationalistic discourse.
It was picked up by cultural historian Arthur Moeller van den Bruck in his eponymous book, Das Dritte Reich. Here he formulated his ideology of the Third Reich as a continuation of the First and Second empires, equating them with the HRE and the German Empire that ended in 1918. His idea was that a synthesis of Nationalism and Socialism would follow the First and Second as the Third Reich in a hegelian culmination of history. A few quotes may serve to illustrate his points:
The German nationalism is the fighter for the end-empire (Endreich). It is always promised. And it is never attained. [...] And it is the special destiny of the German people. [...] There is only one Reich, as there is only one church [...] the German nationalism is fighting for the possible Reich.
The task for German Nationalism would be to pick up the shards of the second Reich, lead it out of slavery and misery into the Third Reich, by the way of revolution. Needless to say that this was immensely popular in the political climate of the Weimar Republic. In that sense, the German Empire of 1918-1933 was viewed as a Zwischenreich, an in-between-Empire, that had no part in the 1-2-3 sheme except as the '-'.
In this revolutionary nationalistic sense, the term was picked up by Nazi propaganda as well, and mingled with the term of a 'thousand year Reich', the eternal Empire at the end of history. 'Drittes Reich' was used in official communication and in propaganda up to 1939.
The big irony here is that 'Third Reich' was not used anymore as a self-description after 1939, its use being officially forbidden by express wishes of Hitler. Instead it was replaced by 'Großdeutsches Reich', the Greater German Empire. Several problems in interpretation had arisen. On the one hand, it was an easy target for anti-regime propaganda that was dreaming of a 'fourth Reich', and extolling the virtues of overcoming the Third. On the other hand, prominent christians argued against the usurpation of christian language and chiliastic concepts by the Nazis, and criticizing the Nazi use of christian terms such as a 'thousand year empire'. In short, the term was an easy target for ridicule and a thorn in the side of the important devout christian demographic. Books and magazines such as 'Die Kunst im Dritten Reich' (Art in the Third Reich) were renamed, and use of the term was discontinued unless in critical circles. 'Thousand Year Reich' especially became an easy target for ridicule as history increasingly began to prove that term a lie.
On a side note, it may be a telling anecdote that Hitler used the term 'Third Reich' again shortly before his end.
Sources used:
Although historians generally term the HRE the first Reich and Imperial Germany as the second, the term is a bit slippery. Although Germans often used "Reich" to refer to both these polities, the the idea of a line of continuity (first, second, third) was much more hazy. The 1923 nationalist book Das Dritte Reich by Arthur Moeller van den Bruck popularized the term "Third Reich" and firmly set the definition of the HRE as the first and Imperial Germany as the second and prophesied that a third and final Reich would complete the work of the other two after a successful right-wing revolution. However, the roots of the term actually predated the Weimar Republic and featured in various nineteenth-century Volkisch works of literature and philosophy. Van der Bruck's book was highly influential among nationalist circles in the Weimar Republic including the nascent NSDAP. After the NSDAP seizure of power in 1933, the official name of the German state was either the Deutsches Reich or the Großdeutsches Reich; the term Third Reich tended to be used unofficially. Hitler actually preferred to use Thousand-Year Reich over Third Reich because the latter implied the regime could be replaced.
The important lesson in all of this is that this process of numbered naming is highly politicized to create a false sense of historical continuity. The HRE had a long and storied history that gets obscured by terming it the First Reich; for Germans the Reich meant very different things in, say, 800, 1648, or 1806. Similarly, van den Bruck's numerical schema deliberately excludes two periods of German history, 1806-1871 and the Weimar Republic, and castigated them as antithetical to his conceptualization of the German polity's true nature.