Melanie provided excellent info above. My personal history with clicker training is this:
I had two rescue dogs before Marlowe, both of whom I trained "the ole fashioned way" and because both were/are tractable, biddable, people-oriented dogs, it kinda sorta worked. They were both "good enough" and because I loathed training them (as did they) because it involved having to be so harsh and mean to them, "good enough" was where we stopped.
Then I rescued Marlowe, a two year old coonhound from a working background. He had never lived in a house before, he'd never been a pet before, he knew no obedience, was not housetrained, and absolutely was lost in this new world he was in. He is not a biddable dog, he was not particularly people-oriented. He latched on to me and my husband more out of desperation than because he naturally wanted to be around people. He's stubborn as an ox, serious as a heart attack, and he doesn't care about my praise, and he doesn't really care about my punishments either. At least, not about the punishments I'm actually willing to dole out. He's not a dog who's super into the social hierarchy either--he's out for himself and himself only and the whole alpha/beta/submissive/dominant game is not part of his wiring. That pretty much took off the table all the forms of training I was already familiar with (punishments-based, and dominance-based).
It took me about two weeks of having Marlowe to realize that I was not going to get through to him without changing up how I was going to go about training in a big way. And I did need to get through to him because he's an adult, 55 lb. dog and he needed to be trained and quickly if he was going to be safe and be able to live in my house without driving everyone else crazy.
Enter clicker training.
I jumped in with both feet and began on my own without a trainer (though Melanie is right, you should get one and I did get one subsequently to help us take it to the next level). The short story is this: Marlowe went from a totally untrained, confused little dog to earning his CGC in 6 months with clicker training. We did the basic stuff (sit, down, loose leash walking, a couple tricks) at home on our own, then took a class to get all the CGC skills we didn't already have (stay, focus, better reliability). Marlowe now is in intermediate level agility class and I may compete with him in the future. He has come an incredibly long way from the dog who didn't even wag his tail when I came home from work or talked to him (for months, he never wagged his tail in response to humans). He's a great partner and friend.
As soon as I began to see the results Marlowe was having with the clicker, I started moving Conrad that direction too, and I was absolutely ashamed and apalled to see the true result of how I'd trained him before. If you've had traditionally trained dogs only, it's hard to see the difference until you get a clicker dog and put them side by side and all of a sudden you see this dramatic difference in their entire demeanor and outlook on life. I saw it almost instantly. Playing the "box game" that Melanie described with Marlowe was a hoot--he was creative and happy and having fun with it. Doing the same with Conrad (at first--he's much better now) was just sad. He wouldn't do anything, he wouldn't move, all he knew was that he'd better sit and not move or mommy might punish him. So that's what he did. In traditional training, you are focusing so much on what you don't want the dog to do. In clicker training your focus should be on getting your dog to do things. To someone who's never had a clicker dog, Conrad sitting and not moving seems like "a good dog". Until you see what a clicker dog does in comparison.