Actually operant conditioning has 4 quadrants. Positive and negative do not mean good and bad. Rather they mean that you add or subtract a stimulus.
Positive reinforcement means you reward, like clicker training.
Negative reinforcement means you remove an aversive stimulus and is the basis of 99% of work under saddle where the horse learns that if he obeys pressure, that pressure stops.
Positive punishment is obvious, but can be mild or severe like walloping a pissy mare that just bit and drew blood out of your butt at the mounting block.
Negative punishment is harder to imagine in a horse context. It means taking something away to punish. Like taking away the car keys from your teen.
One big mistake I see is that people do not understand that in negative reinforcement the learning occurs when the pressure is removed. So I see people continue to spur their horse at every step even when he is going OK. Or they will sing out Good Booooy! while continuing to ride with maximum pressure and think that verbal praise works as positive reinforcement rather than releasing pressure.
Another mistake is when does the intent and effect of negative reinforcement become positive punishment? When you are dealing with a really pissy horse that line can be blurred.
Obviously much that horses do is more complicated than just isolated behaviour. But the behaviourist quadrant does give a good clear schema for thinking about what you are doing in a given moment.
I also think it is really useful for humans to be able to think clearly about their own intent and effect.
Horses already know all about operant conditioning and use it every day in the herd. My mare has even figured out how to give me positive rewards. She is clicker training me to clicker train her.
