The Nightmares are The Passing's boss fights; they are also its dungeons.
Nightmares usually happen upon doing something, or just walking into a certain place. They are distinguished from the real world by the nonsensical architecture (such as a road light stack or lampposts growing out of buildings, or the road going up a house.) and the extremely strange setting.
They can consist but are not limited to fighting a boss, fighting a lot of Stalkers, or resolving a puzzle. Some levels only appear in nightmare versions.
Here's an example of some nightmares:
The Skyscrapers Nightmare
Taking place in two buildings of normally average size, the skyscrapers become overly tall and are linked with each other by bridges. Anything done in one will affect the second one, such as breaking open a vent that leads to a dead end in Building #1 lets you climb into a vent in Building #2 which will let you progress towards the end of the nightmare.
The Museum Nightmare
You will have to find a way to get rid of The Marksmen, a type of Stalker that has the intelligence to use firearms, as you have to run through the now labyrinthic museum.
The Factory Nightmare
Nightmares have no mercy for common sense. In this nightmare, a long run-down factory is completely mixing with a hospital nearby. This nightmare is heaven for the Shockers, as while they appear in other places, are found in masses in here. As such, you will have to rely on your flashlight mostly.