These terms are confusing because most associate the words with value judgments. In training, the terms are usually used in relationship with what is going on in the environment. So a stimulus is either being added or removed and is said to be positive if it is added, or negative if it is removed. Next, remember that we are referring to a relationship between behavior and environmental changes when we refer to reinforcement and punishment. Reinforcement refers to something that results in improvement in conditions for the animal, while punishment refers to a relationship that results in decay or worsening of conditions for the animal.
Negative reinforcement is exhibited through escape or avoidance.
Escape is when something aversive happens and it ends, or the effects are reduced. Avoidance is where the aversive situation is prevented from occurring, or prevented from escalating. So now let's look at punishment. The correct effects of punishment result in the behavior being suppressed, but this can also generalize to something else that was associated with the punishment and can have the same result. If the response or behavior decreases then you know that the punishment was effective.