Earth-0 is the official destination assigned to the universe that includes the mainstream line of publications from DC Comics. It has previously been known as Earth-One and Post-Crisis Earth or the DC Universe. Other names given to include Earth-Cluster and Earth-Sigma. In the present day, the name used varies from the previous mentioned Earth-0, but includes New Earth and Prime Earth.
Earth-0 / Earth-One was created by Gardner Fox and Carmine Infantino.
Note: Earth-One made its first appearance in The Flash #123, but the name Earth-One was not given to it until Justice League of America #21. Earth-0 also was briefly the destination for the Bizarro universe in Infinite Crisis. It was then re-branded as the New 52. In the series Earth-2, it was mentioned that the main Earth from this universe was Prime Earth (Not to be confused with Earth Prime aka the real world). However as most of the multiverse, revealed previous to Flashpoint, was kept intact, the destination Earth-0 remained, which was reinforced by its use in Multiversity and Convergence. Meanwhile the destination Earth-1 was saved for the line of DC OGN and the Bizarro universe was re-branded as Earth-29.
Silver Age / Earth-One
The mainstream universe was called Earth-One in the same event that created Earth-Two and would create the concept of the multiverse in the DC cosmology: in "The Flash of Two Worlds", Barry Allen, then current scarlet speedster would meet the original bearer of the mantle, Jay Garrick. Such discrepancies would be explained by using the parallel earths concept and establishing that the previous characters were from an alternate, secondary world.
From time to time characters from the main universe, now called Earth-One would visit this and many other alternate universes until the death of the multiverse in the epic Crisis on Infinite Earths.
Post-Crisis / Earth-Sigma or Cluster Earth
The Post-Crisis Earth, also know as DC Universe (DCU), Earth-Sigma or Cluster Earth debuted in COIE #11, where it was revealed by Harbinger that all the surviving earths (Earths One, Two, Four, X and S) collapsed and merged their histories into one universe with the previous Earth-One being the most dominant. Elements from the other earths would appear here or there but with the elements of the previous Earth-One being the most relevant. The name Earth-One was dropped and a new continuity was established and the memory of the multiverse was slowly erased from the memory of the heroes, leaving contradictions and unanswered questions.
Many origins were retold and fixed to make it work in the shared reality. Now Superman was not the first superhero, but the first one of a new generation. The JSA still fight in WW2 and the concept of legacy grew strong in the post-Crisis Earth. Sometimes with better results and other with paradoxical or disastrous results as with Hawkman, Power Girl, Donna Troy, The Legion of Superheroes, among the most affected. Such contradictions would be revisited in Zero Hour, where for first time it was revealed the existence of the previous multiverse.
Post-Zero Hour / Earth-Zero
In Zero Hour, the paradoxes from the first crisis were acknowledged. The story was the opportunity for writers to deal with some issues in continuity from the first crisis and made minor changes to adjust the concepts, being the bigger affected the Legion of Super-Heroes. But most of the continuity Post-Crisis stood the same. Elements of fandom also called this new timeline Earth-Zero, in reference to the event and the following consequences, the Zero month, were origins were retold. Continuity would not be an issue again until the 20th celebration of Crisis with another one.
Modern Age / New Earth
With Infinite Crisis, Geoff Johns brought back the previous multiverse, at least briefly, when the main earth was divided again in infinite universes due the machinations of Alex Luthor, a survivor from the previous multiverse. When his plans failed, the forced multiverse collapsed again in one earth, with new minor changes in continuity and the apparent last eradication of the multiverse. The earth resultant was properly called New Earth.
Post-52 Weeks / Earth-0
However, as the weekly 52 revealed the multiverse was replaced with 52 Earths that resemble the previous ones. And being the New Earth the keystone it had been assigned the number 0. Also even if now there was new earths 2, 4, S (now 5) and X (now 10) with their respective characters, there still existed original analogs living in New Earth, as it was shown in the last Justice Society series. However, a last unexpected event would change that situation like lightning.
The New 52 / Prime Earth or Earth-0
After having returning to life in Final Crisis Barry Allen caused a catastrophic change while he fought Profesor Zoom during Flashpoint. When he tried to fix it, a mysterious figure interfered and caused yet another change in the continuity of New Earth that made elements from Wildstorm (then Earth-50) and Vertigo (then, assumed Earth-13) would join the main continuity, but affecting the multiverse too.
After Flashpoint, DC rebooted their universe under title The New 52. The Golden Age elements were mostly lost from the mainstream Earth continuity (Earth-0 / Prime Earth) and rebooted as a new Earth-2. Now on Earth-0, Superman again was the first public superhero (as metahumans, vigilantes and unnatural situations had always happened in secret) and the Justice League was working since five years. Also the visual image of the characters was deeply reinvented, in some cases almost radically. Superman, the most notorious case, loses his iconic red briefs.