Crisis on Infinite Earths: A scientist named Krona once went back in time to observe the beginning of the universe. As a result of his actions, an infinite multiverse was created. One of these universes was composed of anti-matter. In the original universe existed a creature called the Monitor, a sort of embodiment of the multiverse. In the antimatter universe it had a counterpart, the Anti-Monitor, which conquered its universe and began to spread out into other universes. The Anti-Monitor and the Monitor warred for over a million years after they became aware of each other, with the Anti-Monitor conquering and destroying more and more universes as time went by. The Monitor took in a girl called Lyla, the last survivor of one universe, and granted her some of his powers. She became known as the Harbinger.
Eventually the Anti-Monitor's wave of destruction began to encroach on the universes closest to the center of the multiverse, and the Monitor discovered that there were fifteen heroes and villains that together held the key to defeating him. Part of Harbinger's consciousness was sent to recruit them, but at the same time the Anti-Monitor seized control of her body and used her to kill the Monitor, but Harbinger had foreseen the attack and used it as a sacrifice to store two of the last five remaining universes where the Anti-Monitor could never reach them, trusting Harbinger to defeat the Anti-Monitor and stabilize the worlds before they were destroyed by one another's presence. Three other worlds still existed as well and Harbinger and the heroes and villains of those last five earths managed to travel back in time and defeat the Anti-Monitor in his own universe, which caused the five worlds to reset as a single united universe that resembled all of them in certain ways.
During the course of the battle, the Flash of Earth-1 merged with the Speed Force and seemingly died. Only the villain Psycho-Pirate remembered the way things used to be. At the last moment, the Lex Luthor of Earth-3 (a universe where good and evil were reversed) managed to create a pocket universe for himself, the Superman and Lois Lane of Earth-2, and the Superboy of Earth-Prime (a world in which there had never been any superheroes at all until him) to live in.
Zero Hour: After the Cyborg Superman and Mongul destroyed Coast City, the home of Green Lantern Hal Jordan, Hal kind of went nuts and merged with Parallax, the Yellow Fear Entity1 and killed pretty much all of the Green Lanterns and Guardians he could and took their rings. He attempted to use the power of their rings to alter time and bring back Coast City, which had the effect of causing time to start to collapse, starting with the universe disappearing in the distant future and past and closing in on the present. The Justice League managed to stop Parallax, but he had managed to change a few things here and there-- for instance, Joe Chill, the murderer of Batman's parents, escaped justice, and Hawkman's entire continuity got really fucking confusing.
Infinite Crisis: Okay, so remember how at the end of Crisis on Infinite Earths, Earth-3 Lex Luthor (henceforth Alex), Earth-2 Superman (henceforth Kal-L) and Lois, and Earth-Prime Superboy (henceforth Superboy Prime or Prime) escaped to a little pocket universe of their own to live in? Well, from there they were able to keep an eye on the remaining universe, and they weren't thrilled with how things were going. In fact, Prime and Alex started to think that maybe the world they saved wasn't worth saving and plotted to create a new multiverse that would let them find a better earth to replace it with. Around the same time, Lois died, so Kal-L was more focused on that.
Alex had Prime punch through the wall that separated their universe from the main universe. The impact of this caused ripples that screwed with time and changed a few things yet again.
Then Prime and Alex spent a year kidnapping certain important heroes and villains and strapping them into a machine made from the dead Anti-Monitor that would bring back the multiverse as it was before Anti-Monitor started destroying shit so that Alex could merge them into a perfect earth. Prime and Superboy fought, Superboy died, and their battle destroyed the machine, causing all the earths to merge again. Prime went nuts and traveled to the center of the universe (which because of some other complicated shit was no longer Oa) hoping to destroy everything and start from scratch. Superman convinced Kal-L that even if the heroes were darker, the world was still worth saving, pointing to Nightwing as an example of someone who was truly good, and the two of them defeated Prime, but Kal-L did not survive the battle. Eventually Lex Luthor and the Joker tracked down and killed Alex, and Superman, Wonder Woman, and Batman and Robin each took a year to refocus.
Unbeknownst to anyone at the time, 52 universes and the antimatter universe were not merged into the main Earth. Initially these were all the same, but towards the end of that year, Captain Marvel's enemy Mr. Mind metamorphosed into a freaky time-eating being that ate parts of the other universes' timelines, making them all slightly different. Booster Gold, Rip Hunter, and Supernova managed to trap Mr. Mind in a time loop preventing any further destruction, and were initially the only ones to know about the multiverse.
Final Crisis: Hoo boy. If you thought the other crises were confusing... this one was written by Grant Morrison. Suffice to say that there came a time when the New Gods died! Darkseid went for one last hurrah killing his son Orion and finally uncovering the secret of the Anti-Life equation. Ultimately Darkseid was defeated by the combination of Batman sacrificing his life and breaking his code to shoot Darkseid with the bullet that Darkseid used to kill Orion, Barry Allen returning from the Speed Force to lead the Black Racer (the New God of death) to Darkseid, and Superman using a Miracle Machine given to him by Brainiac 5 to gather all his multiversal counterparts to wish for a happy ending, revealing the existence of the multiverse again... yeah, it's Grant Morrison, go figure.
Oh, and there's one other you didn't ask about but bears explaining.
Flashpoint: After Barry Allen came back he found his own history was slightly different-- Eobard Thawne, the reverse-Flash, had gone back in time, murdered Barry's mother, and framed his father. Flash stopped him, but because of wibbly-wobbly timey-wimey stuff this screwed up the timeline immensely. Eventually Barry merged with his time-traveling self and let Thawne kill his mother. While he was moving through time he realized that there were three timelines, and learned that they had been separated to weaken the multiverse against a greater threat, and re-merged them, and thus the new frickin' 52 was born.
1 This wasn't revealed until later on, at the time it was thought Parallax was just Hal being insane.