Oh, Lord. Okay...
The easiest way to break down the DC Universe is with Golden Age, Silver Age, Modern Age, and the newly-minted DC Nu52. The Golden Age started in 1938 with the debut of Action Comics #1, the Silver Age in 1956 with Showcase #4 and the debut of Barry Allen as the Flash, the Modern Age with the 1985 maxi-series Crisis on Infinite Earths, and the Nu52 began last year with the conclusion of Flashpoint. These signposts aren't set in stone, but they make good markers for this discussion.
The Golden Age of DC Comics (then called National Publications) arose in and around WWII. Throughout the 1940's a number of famous characters were introduced including Superman, Batman and Robin, Wonder Woman, Flash (Jay Garrick), Atom (Al Pratt), Hawkman (Carter Hall), Green Lantern (Alan Scott), and many of them joined forces as comics' first super-team, the Justice Society of America.
Around the time the 1950's began, comic books came under attack in the infamous publication, Seduction of the Innocent, and even tied into the McCarthy hearings of the era in some ways. They supposedly promoted amoral behavior like vigilantism, homosexuality, and so on. Lots of other factors were involved, but bottom line is the Comic Code Authority was formed and a major crackdown on super-hero and horror comics began. Practically no super-heroes from the then-fledgling Marvel Comics survived into the 50's, and DC only managed to continue publishing very tame versions of Superman, Batman and Wonder Woman.
Super-heroes burst back onto the scene in 1956 with Barry Allen, the new Flash. Barry Allen supposedly took the Flash identity from the comic book adventures of Jay Garrick, the Flash published in the 1940's. In the Silver Age of DC Comics, then, the Golden Age characters were fictional characters within the current DC Universe going forward. After the success of Barry Allen, DC created other new versions of old heroes, like Hal Jordan, Katar Hol, and Ray Palmer as the new Green Lantern, Hawkman, and Atom. Supes, Bats, and WW continued appearing as if nothing had changed for them, and many Silver Age heroes eventually teamed up as the Justice League of America.
In the genre-defining classic story, The Flash of Two Worlds, Barry Allen and Jay Garrick met for the first time. It turned out Jay Garrick and his Golden Age pals DID exist on a separate Earth (a convoluted explanation was given on how the Silver Age comic book writer who produced the in-universe comic books about the Golden Agers had a psychic connection to that world, which led him to produce comic books about these alternate-dimension heroes...not important.). This story established the Silver Age heroes existed on Earth-1 while the Golden Age heroes existed on Earth-2. The Golden Age heroes were still WWII vets, meaning Earth-2's heroes debuted in the 40's while the Silver Age heroes debuted decades later in the (then-current) 60's. An important note is that while some heroes like the Flash were just two guys with the same name, others like Superman, Batman, and WW were direct counterparts of each other. Kal-El (or Kal-L on Earth-2), Bruce Wayne and Diana existed on both Earths, they were just a few decades older on Earth-2.
The multiple Earths idea proved popular for DC, and for a time they produced a number of different worlds. Earth-3, for instance, was the Reverse-Earth, where Lex Luthor was the planet's sole super-hero fighting an evil JLA called the Crime Syndicate. Most of DC's alternate Earths were actually other companies they purchased alone the way. Captain Marvel (SHAZAM!) was originally published by Fawcett Comics, and when DC bought them out, the SHAZAM! stories became Earth-S. Similarly the Quality Comics and Charleston Comics characters became the heroes native to Earth-X and Earth-4.
DC ran with this multiple Earth concept until they decided it had become too unwieldy and launched Crisis on Infinite Earths to clear up the problem. The end result of Crisis was the destruction of the entire Multiverse, and the history of the universe being relaunched in a single timeline, with only one "New Earth". This literal reboot of continuity merged the history of Earths 1, 2, 4, S, and X into a single timeline. The JLA, JSA, Fawcett, Charleston and Quality heroes now, and always had been, operating on the same Earth. The important thing was that the characters didn't even notice the difference. It took about a year for things to settle in, but after that no one even remembered there used to be an Earth-1 and Earth-2 version of Superman -- there had always been just one Superman. No one remembered the Multiverse had even existed. Pre-Crisis and Post-Crisis became the official watchwords for DC Comics continuity.
DC didn't just throw a bunch of histories into a glass, though...they also turned on the blender. Using Crisis as an excuse, DC lightly or heavily modified the histories of most of their characters, tweaking whatever they thought needed changed in different histories. Superman had a high-profile reboot in The Man of Steel limited series. Pre-Crisis, it seemed like half the planet Krypton survived the destruction and came to Earth: there was Supergirl, Krypto, Zod, the Phantom Zone projector and criminals, the bottle city of Kandor, rainbow flavored Kryptonite, and so forth. Post-Crisis, Superman was the Last Son of Krypton, and all those other characters and concepts were retconed out of existence, including Supergirl's entire history. Superman was also never active as Superboy, but only doned a supersuit as an adult, etc. Wonder Woman was completely rebooted, wiping out her existence in the Silver Age and introducing her to the DC Universe again for the first time in the Modern Age, Post-Crisis. Superman, Batman and WW were all retroactively removed as founders of the JLA, and Black Canary retroactively replaced them to balance the team. The list of Post-Crisis changes goes on and on.
DC was never quite happy with their New Earth, however, and after tweaking history once, they couldn't resist doing it again and again. In 1994 they launched Zero Hour: A Crisis in Time; in 2005 Infinite Crisis, and in 2007 Final Crisis. None of these Crises were as widespread as the original, but they did allow DC to perform minor surgery on their history a few more times. Infinite Crisis in particular was the official sequel to Crisis on Infinite Earths, starring some of the characters who survived the original collapse of the Multiverse, and re-establishing 52 new realities as a minor Multiverse going forward.
The Modern Age of DC Comics was rife with attempts to mold continuity to suit their needs. For example, Power Girl was the Earth-2 version of Supergirl from Earth-1. Although Supergirl and the Superman of Earth-2 were retroactively wiped from existence by Crisis, Power Girl remained. And since DC mandate was that Superman was the Last Son of Krypton, she couldn't be his cousin anymore, even an alternate one since the Multiverse no longer existed. So Power Girl was slapped with a new origin as the timelost granddaughter of an Atlantean sorcerer. This lasted until 2005, when people were allowed to remember the Multiverse again, and Power Girl regained her memories as Earth-2 Superman's cousin.
Presently, DC Comics is setting the ground for another total continuity reboot. The event series Flashpoint radically altered history Age of Apocalypse style. When the Flash "fixed" things, he actually created a new Earth with a brand new history. Major elements so far seem to be the merging of DC Comics with their previously out-of-continuity Vertigo imprint and the Wildstorm titles acquired from Image Comics. Also, Earth-2 has apparently been restored, returning the Golden Age characters to an alternate universe for the first time since the original Crisis. The NuEarth history is still being established, as the Nu52 titles are only about six months off the ground at the moment.
Golden Age: 1938-56. DC is born.
Silver Age: 1956-86. New heroes replace the old. The Golden Age still happened, but on Earth-2.
Modern Age: 1986-2011. Golden and Silver Age histories merge. Many things happened differently than originally depicted.
Nu52: 2011 to present. Many MORE things happened differently. Still working out the kinks. Please buy our comics. Please?