Spider-Man's spider-sense is usually described as a tingling sensation he gets when in danger, non-specific about the nature of the danger but able to identify its direction.
His spider-sense provides an early warning detection system linked with his superhuman kinesthetics, enabling him the ability to evade most any injury, provided he doesn't cognitively override the autonomic reflexes.
Spider-Man wikia says about it's nature:
Several Spider Powered individuals have a "Spider-sense" which is a metacognative ability that warns them of impending danger.
So, we know about how it functions but I don't know about the nature of the spider-sense. Is it a supernatural or psychic sense (as the metacognitive quote seems to imply)? Is it, like the other powers, based on spider genetics (and if so is there any physiological insight about how it works)?
I don't know much about the comics but I guess the explanation can change with time, I'd prefer the current incarnation.