These 4 quadrants are made up two reinforcers (which increase the likelihood of a behaviour occurring again in the future) and of two punishers (which decreases the likelihood of a behaviour occurring again in the future). They can be positive, which in science means a stimuli is added or they can be negative, which means a stimuli is removed. These stimuli are either appetitive, meaning pleasant, desirable things or aversive, meaning unpleasant, undesirable things. Take a minute to look at the infographic to better understand the concept. 

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A trainer using aversive training will discourage undesired behaviour by adding an aversive stimulus (Positive Punishment) and will encourage desired behaviours by removing an aversive stimulus (Negative Reinforcement). So if the trainer using such method wants the horse to perform a specific behaviour he will use something unpleasant enough to cause the horse to want to avoid it.  This method is marketed to horse owner under the euphemistic term "pressure/release". In addition, the removal of the aversive stimulus is often wrongly described as a "reward" by the trainer. This is incorrect as nothing pleasant is being added.

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One thing many horse owners want is their horse to enjoy learning. If you are one of them, then learning how to use positive reinforcement should be a no brainer. Another thing most owners want is to establish strong bonds between them and their horses. The primary function of forming bonds is protection. So how can we expect horses to form a strong bond with humans that they associate with aversives events and stimulus that make them feel unsafe?