The DC Universe (DCU) is the fictional shared universe where the stories in most American comic book titles published by DC Comics take place. DC superheroes such as Batman, Superman, and Wonder Woman are from this universe, and it also contains well known supervillains such as the Joker, Lex Luthor, and Darkseid. It should be noted that in context, "DC Universe" usually refers to the main DC continuity.
The term "DC Multiverse" refers to the collection of all continuities within DC Comics publications. Within the Multiverse, the main DC Universe has gone by many names, but in recent years has been referred to by "Prime Earth" or "Earth 0". It's not to be confused with "Earth Prime".
The fact that DC Comics characters coexisted in the same world was first established in All Star Comics #3 (1940) where several superheroes (who starred in separate stories in the series up to that point) met each other in a group dubbed the Justice Society of America. Subsequently, the Justice Society was reintroduced as the Justice League of America, which was founded with Major League Baseball's National League and American League as inspiration for the name. The comic book that introduced the Justice League was titled The Brave and the Bold However, the majority of National/DC's publications continued to be written with little regard of maintaining continuity with each other for the first few decades.
Over the course of its publishing history, DC has introduced different versions of its characters, sometimes presenting them as if the earlier version had never existed. For example, they introduced new versions of the Flash, which has had loads of characters to take the mantle,Green Lantern, and Hawkman, in the late 1950s, with similar powers but different names and personal histories. Similarly, they had characters such as Batman whose early adventures set in the 1940s could not easily be reconciled with stories featuring a still-youthful man in the 1970s. To explain this, they introduced the idea of the multiverse in Flash #123 (1961) where the Silver Age Flash met his Golden Age counterpart. In addition to allowing the conflicting stories to "co-exist", it allowed the differing versions of characters to meet, and even team up to combat cross-universe threats. The writers gave designations such as "Earth-One", "Earth-Two", and so forth, to certain universes, designations which at times were also used by the characters themselves.Earth-One was the primary world of this publication era.