Definition: {Negative reinforcement} increases a behavior by removing an aversive stimulus. Apply to the case: Taking a painkiller (behavior) $\Rightarrow$ removes pain (aversive) $\Rightarrow$ person is more likely to repeat the behavior. Hence, negative reinforcement.
Why others are wrong:
(B) Positive reinforcement adds a pleasant stimulus to increase behavior (not removal).
(A) Negative punishment removes a pleasant stimulus to {decrease} behavior.
(D) Positive punishment adds an unpleasant stimulus to {decrease} behavior.
Thus, the correct choice is \fbox{(C)}.