Positive reinforcement is the act of enhancing a behavior pattern a rewarding mechanism when the behavior is exhibited. For example clapping or giving a high five each time the desired behavior is exhibited.
Negative reinforcement on the other hand is when a certain stimulus (usually an aversive stimulus) is extinguished after a particular behavior is exhibited. The likelihood of the particular behavior occurring again in the future is increased because of voiding the aversive stimulus. For example, a teacher can ease restrictions from a pupil when he starts to behave well.