In order to understand this, we first have to understand the difference between reinforcement and punishment. Reinforcement is meant to promote a given behavior and make it more likely to occur in the future. By contrast, punishment is meant to make a given behavior less likely to recur in the future. Thus, both attempt to change behavior, but one means to make a given behavior more likely while the other means to make it less likely.
Within each of these, there is a positive way to accomplish the goal and a negative way. The positive way introduces something that was not there before. The negative way takes away something that was there before. Let us look at examples.
A positive punishment introduces an unpleasant stimulus after a given behavior. If your cat scratches a chair and you spray it with water, it is a positive punishment because the spraying was not present before.
A negative punishment takes something good away after a given behavior. For example, if your child misbehaves and you take away their right to use the internet it is a negative punishment. You are trying to prevent the misbehavior by taking away something that had been present.
Positive reinforcement introduces a pleasant stimulus after a given behavior. If a dog performs a trick and you give it food, it is an example of positive reinforcement.
Negative reinforcement removes an unpleasant stimulus after a desired behavior is performed. If you are a coach and you release your team from having to do their daily sprints because they performed well in practice, you are engaging in negative reinforcement.