Note: Long, geeky, and probably boring, plus I undoubtedly screwed something up.]
Back in th 1950’s, DC stopped publishing most of its superhero line -IIRC, only the Superman and Batman titles were published continuously. When DC got back into the superhero line, several of their characters were different versions of an earlier concept. In particular, the original Flash, Jay Garrick (written for most of the original comic’s run by Gardner Fox, was replaced by a different character, Barry Allen. One of the throwaway parts of Barry Allen’s origin was the explanation for his choosing the name “Flash”: he had read a comic book series as a kid, written by a man named Gardner Fox, about a man named Jay Garrick who could run really fast and called himself “the Flash”.
Now. later on, Gardner Fox wanted to write a story that teamed up bothe the Allen and Garrick Flashes. His original idea was simply to have them meet, but this was turned down by editor Julius Schwartz because, he argued, they couldn’t possibly both exist in the same continuity - Barry Allen had read about a fictional Garrick, which wasn’t possible if they were both alive at the same time. Schwartz’s solution, as written by Fox, was to have Allen and Garrick both exist, but in entirely separate universes; in the story in question (Flash of Two Worlds), Barry Allen accidentally crossed between the two Earths by vibrating really fast.
“Flash of Two Worlds” turned out to be really popular, and in consequence a bunch of stories got written on the same premise. In particular, it became a tradition for the JLA to do multi-universe crossover stories with titles like “Crisis on Earth-2” - the latter being a shorthand notation for one of the several Earths that stories could take place on. Earth-1 was Barry Allen’s home, and the JLA’s; Earth-2 had Garrick and the JSA; Earth-3 had the Crime Syndicate (evil versions of the JLA opposed by a heroic Lex Luthor, and there were a bunch of others as well.
Okay, so now move forward to the late 1970’s -early 1980’s. Because of all the different Earths, DC continuity was extremely screwed up - Superman and Batman had been published continuously, so there was no simple explanation of their Earth-1 versus Earth-2 continuity; plenty of comics simply didn’t indicate which continuity they were a part of; and in general, it wasn’t possible for a new reader to understand all of the various facets of the multiple character versions. Consequently, It was decided to completely revamp the continuity of every DC comics property. This resulted in a long miniseries - named Crisis on Infinite Earths, written by Marv Wolfman, and edited by that same Julius Schwartz, which detailed the destruction of the multiverse and the eventual collapse of the remaining universes into a single, consistent continuity. DC also used the oppurtunity to introduce into regular continuity several characters that they owned as a result of buying Charlton and Fawcett Comics some years earlier.
Unfortunately, there were almost immediate problems. The continuity presented in the final issues ofCrisis contradicted the “official” continuity later published as “The History of the DC Universe”, and the Who’s Who series published after that, detailing individual character biographies, didn’t necessarily agree with either. In addition, both Batman and Superman got a retreatment in Post-Crisis continuity by means of a miniseries “recap” of their current origins. In Superman’s case, John Byrne wrote The Man of Steel, mentioned above, as a way of clarifying exactly what Superman’s character was, now. In general, even though there were Superman comics published after the Crisis, it’s generally agreed that Superman’s Post-Crisis continuity didn’t start until the first issue of The Man of Steel. In fact, the last few Superman comics published before Byrne’s takeover were a story by Alan Moore, explicitly finishing the story of the Pre-Crisis Superman.
For characters other than Superman, Post-Crisis continuity is defined by a combination of editorial fiat (DC sez this is how it happened), published origin stories, and Who’s Who references, most of which are way out of date. The earlier History of the DC Universe is generally only a last resort, and the continuity from the last issue of Crisis is definitely ignored. There was a later attempt to “re-fix” some parts of continuity, via the Zero Hour miniseries, but it didn’t really work, and it was furthermore undermined by the introduction of the “Hypertime” concept, which was basically Multiple Earths in disguise. Luckily, Hypertime itself is also currently generally ignored.
As far as the movies go, that’s considered separate continuity. Superman as presented in the first two movies is based on Pre-Crisis contiuity, because they were made long before Crisis happened, however, the movie Superman isn’t necessarily presented the same as the comic Superman, and the events of the movies aren’t supposed to have happened to the Superman of the comics, though Superman II was based on an earlier comic story.