Operant conditioning means that a behaviour becomes more or less likely as a result of its consequences.

A punishment is something that makes a behaviour less likely to happen, and a reinforcement is something that makes something more likely to happen. 

The terms positive and negative are here used to describe adding or removing something, rather than good or bad.

Positive reinforcement means adding something that the horse likes to produce a behavior that we want, like giving a food reward or a scratch when the horse does the right thing. 

Negative reinforcement means removing something the horse doesn’t like to make a behaviour more likely, like removing a leg aid when the horse moves forward. 

Positive punishment is adding something the horse doesn’t want, to make behavior go away, like smacking a horse that kicks. Most of the time, this is a method we should not use since it’s very hard to do correctly, and it doesn’t teach the horse what it should be doing. 

Negative punishment means ignoring a behavior that we don’t want. 

Combined reinforcement is using negative and positive reinforcement together - you take your leg off when the horse moves forward, and also reward it. This is often the most effective way to train a behavior.