According to the legends, they cast mankind out from the Tranquiline Halls to Roshar and, from there, they tried to force mankind out of Roshar to Damnation. As well, according to the legends, they were defeated during Aharietiam.
Szeth mentions that, according to legend, Voidbringers were "horrors of rock and flame, dozens of feet tall, foes whose eyes burned with hatred." These legends most likely refer to Thunderclasts, which are possibly a type of Voidbringer. They might be Parshendi bonded with Voidspren or stormspren. There are probably ten types of Voidbringers, just as there are ten Orders of Knights Radiant.
In Words of Radiance, it is revealed that the Voidbringers are, in fact, the Parshendi when they assume stormform, characterized by their red eyes. The parshmen can also transform into Voidbringers, though they lack the ability (or initiative) to transform on their own. The Everstorm carries spren that will turn Parshmen into Voidbringers on contact. 
Voidbringers are created when certain creatures, such as Parshendi, bond with spren of Odium (i.e., stormspren).
According to Kalami, the Voidbringers came to annihilate. She tells Dalinar that their goal was to wipe humankind from Roshar. She elaborates by saying that they were specters, formless ... that some say they are spirits of the dead, others spren from Damnation.
|“||"Voidbringers have no pattern. Besides, I have read of them in your lore. They speak of spindly arms like bone, and horrific faces. I should think, if you wish to find one, the mirror might be a location where you can begin your search."||”|
|“||"They were creatures of terrible destructive power, forged in Damnation, created from hate."||”|
|“||"Everything has its opposite, Shallan. The Almighty is a force of good. To balance his goodness, the Cosmere needed the Voidbringers as his opposite."||”|