One way to think about training is to figure out for yourself what kind of person you want to be to others; human, canine, porpoise or lion. The principles of learning theory generally hold across species, it's the reaction that might be different.
Lets say we want to bully a human around (likely Positive Punishment). The reaction might be to leave, cry in a bundle on the ground, or to call the Principle/boss or get a divorce. A dog, depending on it's temperament might fight back, or it might cower in submission trying to end the unpleasantness. A porpoise will simply swim away. A lion might just eat you.
Nagging can work with a person (Negative Reinforcement), but it rarely works with any other species. You can't nag a dog into sitting "Sit... Sit... Sit... SIT... SIT!! SIT!!!!" If you've gotten to that point, go back to basics and teach the dog to sit (see the second post :) )
Removing something a person wants can be useful, if they understand what it's being removed for (Negative Punishment). Stopping and telling someone that did something wrong 3 years ago and now you don't want to be with them anymore is somewhat pointless... even more so is not telling them why at all. One example that might work on a child is they do something wrong (screaming and yelling at you) then immediately taking the XBox/PS3 etc and putting it out of reach; assuming the child cared about it at that moment. A jumping up puppy can often be taught not to jump by simply turning and ignoring it; a dog that has been reinforced for jumping up though, may continue to do this for a long time with this technique before it figures out what's going on.
If I gave you a cookie (or piece of chocolate) every time you read over one of these posts (and you liked cookies or chocolate, or wine) then you would likely come back and read more... or maybe even the same post over and over (Positive reinforcement). Reward a dog for sitting ever time it comes running up, and it is more likely to happen. Give a porpoise a piece of fish ever time it jumps through a hoop, more likely to happen. This is again, contingent on if it's something the animal actually likes at the moment!. A porpoise coming up to you might just want a stroke, and so might that dog.
So, completely confusing? Negative == removing something; Positive == adding something. Reinforcement == makes the behaviour increase in likelihood; Punishment == decreases the likelihood of the behaviour. Still likely clear as mud.
Positive Reinforcement. Probably the easiest one to figure out. Get a reward for doing something, you want to do something more often (getting paid at the end of the week is deferred, so it doesn't work the same way though, sometimes even makes the thing happen less!). Get a hug for taking out the trash (assuming you like hugs!), you might feel better about taking the garbage out. Ask a dog to sit and he gets something he likes in return - increases his likelihood of sitting.
Negative Punishment. Somewhat easy to think about; when you were a child, and you acted up and something you were playing with got taken away, you realize it's a bad thing to act up; your fun stops. Similar to a puppy jumping up, or nipping to get attention. If the attention immediately evaporates, then the game is over, fun ends. If I bark my head off while meals are being prepared, my person leaves, and I don't get food... hmmm Slightly tricker to figure out, but the pup gets the pattern quickly if you're timing is spot on.
Negative Reinforcement. Trickier to explain. Say the pup doesn't like pressure on its rear end. You put pressure on it's rear, and it moves it's butt to the ground to get away from the pressure, and the pressure stops! Voila, the puppy sits, but who did all the work? You. If you did all the work, how fast is the puppy really going to learn? Slowly. Negative reinforcement isn't always just applying something annoying unfortunately; a lot of people use things that hurt or are at best uncomfortable to manipulate animals. If it's something you wouldn't use on you, your child or other loved one, don't use it on a dog or other animal.
Positive Punishment. Adding something to prevent the things you don't want. The classic example of this in old dog training was yanking a chain until the dog sat. Everything it did wrong it got a yank, until it eventually sat. Not a lot of fun, and the dog has to figure out what not to do! Not the ideal learning condition; think about learning the alphabet this way.
Why did praise (sometimes Positive Reinforcement) work so well in the old days? Because it was one small piece of a larger puzzle. Mostly methods that espouse "praise based" training rely on Positive Punishment/Negative Reinforcement, followed by praise. So dogs likely assumed that if they did what they got praise for, then the bad stuff wouldn't happen. Unfortunately, this can still be seen on TV, even though the techniques are 50 years out of date and probably only came into favour after the World Wars.
Refer to Ian Dunbar for his history on rediscovering lost books from the 1800's that described the basic concepts of Positive Reinforcement and Negative Punishment together in animals before the scientists coined the terms with Pavlov et al.
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