Reinforcement: Adding or removing something to increase the likelihood of a behavior recurring.
Positive – Adding something pleasant/desirable, like giving a dog a treat for sitting when you ask her to. When the dog sits, she is reinforced by receiving a treat.
Negative – Decreasing something that is unpleasant/undesirable, like releasing tension on a collar when a dog walks calmly at your side. When the dog walks calmly, she is reinforced by no longer having tension on the leash.
Punishment: Adding or removing something to decrease the likelihood of a behavior recurring.
Positive – Adding something that is unpleasant/undesirable, like applying pressure to a head collar when a dog pulls or lunges. When the dog pulls, he experiences pressure around his snout similar to the correction dogs give one another when they play too roughly.
Negative – Decreasing something that is pleasant/desirable, like withholding your dog’s food bowl until he waits calmly for it. When the dog barks or lunges for his food, he doesn’t get fed.
Of the above, negative reinforcement is the trickiest to understand, I think, and also the one least likely to show up naturally (without following positive punishment). Increasingly, trainers are moving toward methods that motivate the dog to engage in desirable behavior, noticing the behavior they want and reinforcing that.
Whenever possible, I work with positive or motivational training, and in the rare cases when I’m trying to extinguish a behavior that hasn’t responded to other strategies, I may add positive punishment and with a lot of forethought. For example, I do use head collars, and I make a soft hissing noise when my dog barks excitedly for extended periods of time. Her initial startle reaction allows me to redirect her to a more desirable behavior that I can reward instead.
I wouldn't expect every person working with his or her own family dog(s) to fully grasp all of these concepts, but I do think that grappling with them, engaging in thoughtful conversations, and analyzing your own training methods and beliefs will help you make informed decisions when engaging with desirable and undesirable behaviors in your own home.