Nociceptive
Normal "tissue damage" pain. Somatic pain comes from skin, muscle, or bone. Visceral pain comes from organs, and it may refer elsewhere. For example, a fracture or pancreatitis.
Neuropathic
Pain from nerve injury itself, in the PNS or CNS. It burns, shoots, or tingles. For example, diabetic neuropathy, shingles, carpal tunnel, or Guillain-Barré.