Damnation is the equivalent of "hell," the concept of everlasting divine punishment and/or disgrace, and especially the punishment for sin as threatened by the Almighty in Vorinism.
According to Vorin theology, during Desolations, the Voidbringers came again and again, trying to force humankind off Roshar and into Damnation ... just as they had once forced humankind and the Heralds out of the Tranquiline Halls, the equivalent of "paradise". Damnation was described as the place where the Voidbringers were forged from hatred.
Damnation likely refers to the place where the Heralds spend time between Desolations, a place of eternal fires and torture by the souls of the vengeful Dawnsingers, slaughtered by humans when they first came to Roshar in order to keep them contained and from reincarnating. (This would explain why the Herald Talenel appears so exhausted when he returns to Roshar at the end of The Way of Kings.) The herald Nin told Szeth that Damnation is located on the planet Braize.
"Damnation" is used throughout The Way of Kings by all characters, as a swear or curse, such as "Storming Damnation","There's no way in Damnation" or "Damnation's own fool!".
The concept of Damnation is likely shared by peoples outside those countries adhering to Vorinism; it was mentioned by Szeth, despite his being from Shinovar. He believed that his murders and sins would damn his soul to a continued existence of agony and punishment.
"The flesh burning. The fires. The pain over and over ... ."