Positive Reinforcement is proving someone with a positive outcome when they engage in a behavior you consider desirable in order to increase the frequency with which they engage in the behavior.

Negative Reinforcement is giving someone a negative outcome which the person will seek to avoid by decreasing the frequency of a behavior you consider undesirable.  

Some of the most important principles of teaching follow:

1) Although negative enforcement sometimes works, no student every learns without receiving some positive reinforcement, because the student does not know which behavior is better.  Without positive reinforcement, no learning occurs.

2) Every student has his or her own ideal ratio of positive and negative reinforcement which will maximize learning.  

3) Even if negative reinforcement helps in a particular instance, the teacher needs to monitor closely whether the negative reinforcement is demoralizing the student, which will result in a long term decreased rate of learning.  This is called "punishment" or "learned helplessness".

The brain chemistry of the student actually changes when the student receives positive reinforcement.  Each instance of sincere positive reinforcement results in a slightly improved brain chemistry of the student.  Each instance of negative reinforcement, regardless of degree of negativity, results in less improved brain capacity.  After receiving just one instance of negative reinforcement, a student requires approximately 10 instances of positive reinforcement just to return the brain to its state prior to the negative reinforcement.

The trick is... to catch the person doing the right thing... and then provide sincere praise...