Negative Reinforcement is taking away something to increase the likelihood of the behavior happening in the future.
An aversive stimulus is removed when the dog does the desired behavior making that behavior more likely to happen in the future. Negative reinforcement makes it more likely that the dog does the behavior. It is not punishment.
This is often perceived as a truly nasty training technique. An ear pinch for a hold, for instance: the handler applies an ear pinch and removes the aversive stimulus when the dog bites and holds the desired object. It is not a pleasant training technique.
But there are many grades of Negative Reinforcement. The Dead Fish Tug can be Negative Reinforcement. Freezing and waiting (aversive) while the dog is not doing what you want only to spring back into action when he complies (removal of aversive). Not getting cookies can serve as Negative Reinforcement as well.