There are 2 main types of fibers (i.e. neurons) which carry painful stimuli up the spinal cord to the brain, and each carries its own specific type of stimuli.
A-Fibers: carry sharp sudden pain (touching a burning stove, stubbing a toe, hitting a funny bone). These pathways mediate reactive pain which instantly tells the brain that something is wrong and allows a protective reflex to kick in. For instance when you lay your hand on a scalding hot stove, crush your fingers in the car door, stick a fork in an electrical outlet, get kicked in the nuts, or when your girlfriend gets angry and slices your dick off with a box cutter in your sleep - the pain triggers a reflex telling you to pull away, or to scream and cover what's left of your dick with your hands.
Notice that morphine doesn't necessarilly make you immune to sudden reactive pain, but rather dulls any current underlying pain. The closest one will get to "immunity" from sudden pain would be with a high dose of an ultra-potent opioid such as fentanyl. There are actually anecdotal reports of individuals who begin to experience a loss of noxious sensation with acute high doses of fentanyl - indeed, high doses of fentanyl type drugs by themselves can produce complete general anaesthesia (i.e. a full blockage of pain transduction at the dorsal horn). However, anyone taking a high enough dose of fentanyl to experience this is lucky to be alive - No one outside of a hospital-surgical setting should be dosing anywhere near high enough to lose sensation (i.e. "induce anaesthesia"). That's right; anaesthetizing yourself at home is a bad idea.
C-Fibers: carry dull constant pain (headache or toothache, cancer or illness, chronic pain)
Opioids work by targeting ascending C-fibers. They are therefore effective in relieving deep, aching pain - this includes the pain that is felt in the days or weeks following injuries or surgeries, as well as chronic pain experienced following severe injuries or occurring due to a range of conditions.