Dog training collars come in all shapes and sizes and for all breeds and ages of dog. You fit a training collar around your dogs neck (assuming they're OK with that and don't immediately pull it off), and then it is either remote controlled or has a sound activation.
Some "dog training" collars are actually just ordinary collars, and the makers will encourage you to attach a lead to them and give it a yank when the dog misbehaves. I disagree with this approach to dog training, partly because it's dangerous and partly because it generally doesn't do much good unless done perfectly.
Of the other types of dog training collar, they fit into two broad categories: anti-bark and general purpose.
The anti-bark dog training collars are sound activated. They have a small box of unpleasant smelling perfume just under the dogs nose, and a little sensor which detects when the dog barks. As he (or she) barks, a small squirt of the perfume is released, irritating the dog. The idea is that eventually the dog will begin to associate the unpleasant smell with barking, and so stop barking.
While this device has it's roots based in the psychological bell ringing dog studies conducted by Pavlov, general consensus amongst dog trainers is that these dog training collars are to be avoided. Good training regimes should reward dogs for good behavior instead of punishing bad behaviour.
The other type of collar is very similar, except instead of releasing bad smelling perfume, these training collars may impart a small electric shock to the dog wearing it when you press a button on a remote control. The advice given by the manufacturers is that you can discipline your dog from afar, if he's about to chew up someone else's football, ruin some plants or is running around a kids playground barking his head off and you can't catch him.
Again, I'd recommend that you avoid these dog training collars. Electrocuting your dog is not the way to get him to behave, and any collars that you put on him (or her) could just make him more agitated and excitable, and exacerbate his behavior, not control it. Imagine if you were angry and letting off some steam, and someone started shocking you? You wouldn't be very happy, would you?
The best approach to dog training isn't collars, it's a full and complete training system developed by professionals, which encourages a strong relationship between you and the dog, and rewards for good behavior. There's plenty of professional advice available so you don't have to resort to dog training collars.