The Eternal Conflict is the name for the war that has raged on since the dawn of time between the angels of the High Heavens and the demons of the Burning Hells over the control of the Worldstone at the heart of Pandemonium. This war raged for untold aeons, with one side controlling the Worldstone for a time until the other side gained control of it.
Eventually the archangel Inarius grew tired of this endless conflict and decided to leave it. He found like-minded angels and demons, and together with the demoness Lilith , stole the Worldstone from the Pandemonium Fortress by warping it to another dimension and with it created the world of Sanctuary.
Utterly surprised upon the discovery of the Worldstone missing, neither Heaven nor Hell knew what had happened. Since the heart of the conflict stood around the Worldstone, both sides initially suspected the other of treachery, but it soon became obvious that the stone was missing, and so both sides began a frantic search for it. It was once again found when the Sin War was ending and Heaven discovered Sanctuary. Both Heaven and Hell realized the potential that mankind held, and thus made a truce and a non-interference pact, that they would let humans choose freely to ally themselves with either of them.
The Three almost immediately began to scheme in order to circumvent the pact, something the Lesser Evils did not like as it seemed to them the three brothers were hesitant to push through with the war. As such, a civil war broke out in Hell between the Prime Evils and the Lesser evils that raged for a long time. The Three eventually lost as even their own demons turned against them, and this led to the Dark Exile wherein the Prime Evils were exiled to Sanctuary. Meanwhile the truce between Heaven and Hell seems to have been maintained.
It is unclear why the Three were exiled to Sanctuary of all places, though this seems to have been intentional if we are to believe Izual who claims that the Three masterminded their own exile to Sanctuary. Perhaps this was done in order to maintain the truce with Heaven, thus leaving the Three free to roam Sanctuary without risking direct interference from either the High Heavens or from Azmodan or Belial.
It is also unknown why Azmodan came to be the ruler of Hell and what happened between him and Belial after the Three had been exiled, whether the two still worked together or if Azmodan after the Dark Exile turned on Belial and defeated him as well.
 Previous Names
The Eternal Conflict have previously been called both the Great Conflict and the Sin War. The former is no longer used, and the latter now refers to a specific part of the Eternal Conflict which is told in the Sin War novels.
On Level 4 in Diablo I, a tome titled "The Great Conflict," surrounded by sconces, will voice you the following as soon as you open it:
"Take heed and bear witness to the truths that lie herein, for they are the last legacy of the Horadrim. There is a war that rages on even now, beyond the fields that we know--between the utopian kingdoms of the High Heavens and the chaotic pits of the Burning Hells. This war is known as the Great Conflict, and it has raged and burned longer than any of the stars in the sky. Neither side ever gains sway for long as the forces of light and darkness constantly vie for control over all creation."