Positive reinforcement works by presenting a motivating/reinforcing stimulus to the person after the desired behavior is exhibited, making the behavior more likely to happen in the future.
Negative reinforcement occurs when a certain stimulus (usually an aversive stimulus) is removed after a particular behavior is exhibited. The likelihood of the particular behavior occurring again in the future is increased because of removing/avoiding the negative consequence.
Negative reinforcement should not be thought of as a punishment procedure. With negative reinforcement, you are increasing a behavior, whereas with punishment, you are decreasing a behavior.