Something that's Stygian is dark, murky, and probably a little melancholy. Your walk home from the bus stop might feel Stygian on a foggy, moonless night.
When you describe something as Stygian, you're comparing it to the murky and terrible river Styx, which flows through the underworld in ancient Greek mythology. The word itself comes from the Greek word Stygios, from Styx, which literally means "the hateful." Some related words in Greek are stygos, "hatred," and stygnos, "gloomy." That should give you a good idea of the way a Stygian library, alley, or cave looks and feels — dismal and dark.