I like my tl;dr too much.
Contrary to whatever wild speculations and beliefs fly around in this time and age, the Earth was in fact not created millions of years ago - rather, it was created in the year 4004 BC, Sunday the 21st of October sometime around 8.45AM or so because apparently God just liked to do that and nobody can really refute him since he is, you know, God.
Anyway, to the story at hand. Crowley (once known as Crawly, the infamous Serpent of Eden) and Aziraphale (a Principality, and the angel of the Eastern gate) are respctively Hell and Heaven's agents in the world, charged with the task of living with the humans and tempting/thwarting them as such. That was a task the duo did most zealously in the first couple of millenniums or so - its really hard to say just exactly how many fights and skirmishes the two had against each other in line of defending their mission or whatever it was supposed to be or how many times they had discorporated each other through their fights. But whatever the case, one can only try and kill--er, discorporate each other for so long as it were, and eventually the two came to the awkward realization that they had much more in common with each other than their immediate (but very far off) bosses. And through that realization, eventually in the year 1020 the two came to their Arrangement.
The Arrangement, by all means and ways, was really a very basic thing. So basic and open-ended in-fact, that it only got the capital A for having existed so long. The Arrangement was basically a non-interference clause, meaning to say that it was a way to make sure that while neither side won, neither side also lost - balancing the two sides they worked for respectively, to put it simply. It also got them out of the constant need to get a new body (because the bureaucracy of Heaven and Hell were both equally horrible when it came to that) and more importantly, it got them of the need to constantly face their bosses which was probably the best part.
So with the Arrangement in place, the two entities lived comfortably for a good long while until the year 1990, where it seemed that Heaven and Hell finally decided to take things for the final showdown and planned to start Armageddon, much to the duo's discomfort. For after having stayed so long on Earth and having watched all of humankind (who had fallen from His grace because of the whole Adam and Eve thing, courtesy of Crowley) progress from their humble start, both Crowley and Aziraphale decide that they liked Earth and humanity, and decided to put an end to the Antichrist and not let Armageddon happen. Unfortunately though, a few things happened during the night Crowley was supposed to make the switch of the baby with the Antichrist, and the boy the two had watched over for the next eleven years turned out to not be the Antichrist at all.
Thus the search for the real Antichrist began, and after sort of accidentally stumbling upon the Nice and Accurate Prophecies of Agnes Nutter, Witch both Crowley and Aziraphale discovered the true identity of the Antichrist - Adam Young, a young and somewhat charismatic eleven-year-old living in Lower Tadfield and had already named the satanic hellhound given to him Dog. A whole bunch of things happened though, like Aziraphale's bookshop being burned down (Crowley was very distressed about this, though he will never admit it), and Crowley's own car getting on fire when he drove like hell (pun unintentional) towards where the Apocalypse was going to take place: Lower Tadfield Airbase. It almost seemed like all hope was lost as the Adam had arrived as well with the four Horsepeople of the Apocalypse, but then the kid decided that he liked Earth too and pretty much just called it off.
One would think that would be the end, but both Heaven and Hell didn't seem to be happy about that and tried to start the Apocalypse anyway - both Crowley and Aziraphale turned into their true forms in a bid to try and protect Earth by themselves... but then Adam simply invoked his Antichrist powers and returned all of Earth back to normal before anything had ever happened.
...so it was like this was the end of the world averted, and ever since then both Crowley and Aziraphale lived their lives comfortably on Earth as always, cleaning up after all the hoo-haa that was the Near-Apocalypse and seeing humanity continuing to live and strive despite their mortality and frailness. Heaven and Hell both aren't the most pleased with them, but they still do their jobs as told (or as much as they can anyway) so they're left alone for the most part - which is exactly what they wanted.