Real world stuff:
From the mid 60s, until 1986, DC comics had a multiverse - the main characters lived on Earth-1, the Golden Age characters on Earth-2, evil versions on Earth-3, Captain Marvel on Earth-S, the Nazis won WWII on Earth-X, etc.
In the 80s, they were flagging behind Marvel in sales, and decided that part of it was the Multiverse making it hard for new readers. (Whether this is the case or not is debatable, but DC believed it at the time.) Crisis (named for many of the crossovers using the Multiverse, which were often titled Crisis on Earth-Whatever) was an attempt to clean that up without just saying ‘we’re starting over’ and chucking everything.
Plot stuff:
The story of CoIE, simply stated, involved a villain named the Anti-Monitor attempting to destroy all of the matter universes and incorporate them into the anti-matter universe. His opposite number, the Monitor, recruits heroes and villains from multiple universes to prevent him from succeeding. Though Anti-Monitor recruited allies of his own - specifically the Psycho Pirate.
This involved two things - combining the five universes which hadn’t yet been destroyed (1, 2, 4 (home of the recently acquired Charlton characters), S and X) into one, which could be protected from the anti-Monitor, and, of course, defeating the anti-Monitor. The first task involves teams sent to various periods in time (the age of Atlantis, the old west, Kamandi’s future, the 30th century), to set up ‘tuning forks’ to allow the merger.
Both tasks were successful, but the Monitor’s group suffered casualties (including the Monitor himself) - Supergirl of Earth-1 and Barry Allen, the Silver Age Flash being the most famous.
In the end, there was one unified universe, but several characters - Superman and Lois Lane of Earth-2, Superboy of Earth-Prime and Alexander Luthor of Earth-3, found themselves with no place in this world, so they sealed themselves in an alternate dimension they perceived as heaven. (This would later be revealed to have been a major mistake on their part, but not for 20 years.)
Eventually, only one man, the Psycho Pirate, remembered the Multiverse.
Back to the real world:
It actually took some time for the post-Crisis universe to gel - Superman, who got the most thorough reboot, had a whole year before they stopped doing stories clearly (in hindsight) set on Earth-1, and other characters - Hawkman, Power Girl, the Legion of Super-heroes being the most glaring examples - would go for a decade or more before the wrinkles were ironed out. (It was so bad for Hawkman, he was declared ‘radioactive’ for a few years.)
But, the post-Crisis universe was as coherent as super-hero universes ever are, and made the universe more or less what DC wanted it to be at the time (jettisoning the sillier aspects of the Silver Age, and the multiverse, while maintaining some continuity with what came before), so, it was, at least at first, a success. And, hey, it gave them an excuse to retell some classic stories.