Okay. Here's how it breaks down. Kirby left Marvel for DC because DC and he took the New Gods idea with him (I knew about it being a spiritual successor to Thor but didn't recognize he'd actually planned it like that. That's cool, Proj). The Fourth World was ostensibly going to be a separate entity from the DC superhero universe proper, a big cosmic mythological tale with characters he'd own the rights to and have editorial control over. DC gave him credits for the characters and let him basically edit. He put out four books, with three being main series and Jimmy Olsen basically using a crossover character to introduce the cosmology to new readers. It was, essentially, something like Thor, but without the constraints of being too deeply tied into what was running in the core superhero books at the time. The books got canceled before the story wrapped and the characters got folded into the DC universe proper, where they appeared infrequently and treated mostly as superhero characters who happened to be gods (again, see Thor), but were really just treated as space men with goofy technology. Darkseid apparently had a few good spots as a villain. There was a big Legion of Superheroes story where he was the bad guy that was apparently pretty great, though I haven't read it. He basically got folded into Superman's rogue's gallery, largely I'd assume, because he was one of a few characters who could be a credible physical threat to Supes.
The nomenclature of Fourth World basically came down to god cycles, the idea that there'd be gods before and gods after. See again Proj's Thor comment above. The gods themselves were supposed to reflect the modern age, with Darkseid and Co.reflecting the values of post-WWII mechanized warfare and the good guys representing, Idunno, the human spirit or something. Supposedly Kirby intended it to be a finite story with a beginning, middle, and end, concluding with the fall of the Fourth World. At the center of it was a conflict between Since then, they introduced Gog (from Kingdom Come) as a god of the Third World in the JSA book, which would suggest some of these elder gods might still be around. Which brings us around to Morrison.
He resolved the war in heaven, with the New Gods all ending up dead. The premise of Final Crisis was that Darkseid was the last one alive, but he was fatally wounded and rather than letting a new generation of gods take his throne, he'd rather take all of reality down with him. And here's where it gets dicey. In the process, he sent Bruce Wayne back in time with a device that turned him into a living, reality destroying weapon. If you read Final Crisis you'll recall a minor bit of window dressing involving the idea that pieces of ancient graffiti were being found representing the symbols of Superman, Wonder Woman, and Batman. As Bruce traveled through time, he sketched these symbols, implanting them in the human subconscious in the process, rewriting time so that, in this new chronology, the symbols of these heroes are totemized. So in effect, all of reality has been rewritten with the symbolism of the heroic age implanted in the human psyche as far back as prehistoric times. So, by sending Batman back, Darkseid effectively set the stage for the Fifth Age, which is basically "man as god", the potential of common man to rewrite himself as something divine. Batman is presumably the soldier god, Super Young Team are the new Forever People, and other characters probably fill unrevealed spots. It's a cute conceit, but one that's so meta that it's rather hard to fit in to most storylines, and given that even Morrison's mundane ideas are usually left alone by other creators, it may be something that never really shows up in continuity again.
Oh, and in the new multiverse, apparently the Fourth World gods have reborn in a Kirby-universe that contains Kamandi and all of Kirby's other creations. I'm not sure what that's about.
So, to summarize... The Third World are primitive (maybe Old Testament?) gods, the Fourth World are modern gods, and the Fifth World are post-modern gods.