The 4 quadrants of conditioning - courtesy of BF Skinner -
Positive Reinforcement (+R): If you want your dog to repeat a behavior more frequently, GIVE the dog something, reward that behavior in some way.
Cookies for sitting.
Negative Reinforcement (-R): REMOVING an aversive (something unpleasant) when the dog does the desired behavior. It can also mean withholding the delivery of an aversive to reward the dog for performing correctly. The dog is rewarded by escaping or avoiding the aversive. Most commonly in dogs, this is used to teach traditional force fetching - it's an ear pinch until the dog opens his mouth and takes the dummy.
Negative Punishment (-P): If you want your dog to repeat a behavior less frequently, REMOVE any reward or perceived award for the behavior.
This is the one that is hardest to wrap your mind around and is frequently misunderstood, even by the best dog trainers. So, you want your dog to do something (like pay attention in the agility ring) - if the dog zooms - which is rewarding - you punish the dog by removing him from the ring, sticking him in his crate and ignoring him for a set amount of time. This is known as the walk of shame and is sometimes useful with dogs who find the game of agility very rewarding. Other dogs never make the connection. I have a friend who is an EXCELLENT trainer who swears this doesn't work. However, I had one dog (out of 4 I've trained in agility) who it worked on quite well. No play, no pay.
Positive Punishment (+P) - anything the trainer GIVES immediately following a behavior that tends to decrease the likelihood of the behavior being repeated - for instance rattlesnake aversion training - a shock from a collar when the dog approaches a snake. Or less extremely, a leash pop when a dog starts to pick up trash.
Truthfully, most trainers combine all of these. You rarely find a completely +R trainer, simply because sometimes you throw in some -P in there. So let's say you guide the dog to sit with a cookie. At some point, you've got a cookie in your hand, the dog will not stay in a sit and you remove the cookie from sight. You've increased your criteria for that cookie. That brief bit of -P thrown in reinforced the +R.
You WILL find trainers who will avoid -R or +P under any circumstances. There are ways to train hunting dogs without a shock collar or obedience retrieves without a force fetch. It may take longer and more patience, but it can be done.