There are two main classifications of pain: the commonsensical sort that arises from damaged tissue (nociceptive pain), and the more exotic kind that comes from damage to the system that reports and interprets damage, the nervous system (neuropathic pain). This is the difference between engine trouble and trouble with that light on your dashboard that claims there’s engine trouble. Oddly, there is still no official “other” category for the pain of conditions like fibromyalgia and irritable bowel syndrome, which involve dysfunction of the nervous system, as opposed to damage; names like nocipathic or algopathic are on the table.
Nociception and pain are not equivalent and there are no “pain fibers,” just nerves that send data to the brain for consideration. All pain is technically a brain-generated experience. However, the illusion that pain is “in” our body is meaningful and functional.
Pain can also be classified as somatic (skin, muscle, bones, joints) and visceral (organs).