The Dark Voice is a dick, to put it simply. But he's also an insane dick. He whispers in people's heads and drives them mad. He also wants to remake the whole of life in his image for some reason.
The Overmind was possessed by the Dark Voice, giving it an overriding directive to assimilate the Protoss. He wanted the Zerg and Protoss to merge to create creepy Hybrids, which apparently look and act like him. He would then use these Hybrids to wipe out the remaining Protoss and Zerg.
The Overmind clocked onto this plan early on and tried to work against it, selecting a suitable leader to take over the Zerg Swarm and free it from the Dark Voice's control.
In the end, the Dark Voice's plan failed, with the Zerg being freed and joining The League, and he apparently fled. His location is still unknown.
Rumours circulate that the Dark Voice now stalks League Territory, getting his kicks by creeping people out and hiding under children's beds. Next time you venture down a dark alley alone, beware the darkness...