I'm just going to give my personal experience with clicker training, so take it or leave it.
I love clicker training. It was the only thing that really worked on my dog. She's a chow chow, they are known for being very loyal loving dogs that will protect and look after their owners until the end. However, they are not known for their obedience. We went through a very popular trainer in the area that everyone said was great. He did use choke chains and leash corrections in his training and also was very against using food as a motivator. Well, 6 weeks of training classes and 2 private lessons letter I had a chow chow who would still die for me, but not sit for me
A friend of mine showed me how she trained for dog, a 10 year old sheltie who had her Utility title in obedience, using the clicker. The click is used to as a marker to pin point the exact moment the animal does something right and communicates that very clearly to the animal as well as tell them that they are going to get rewarded for that (the click it self is not a reward, food is used a lot because most animals are motivated by food, anything can be a reward but it HAS to be something the animal likes it)
So, because the click is able to pin point the exact moment when an animal does something you like, you are able to break down a larger behavior into very small steps that can easily be accomplished by the animal (your setting them up for success). Animals and also humans learn a lot quicker and retain information a lot long when tasks are taught to them in small steps (think what when you get a new job, is it easier if the show you step by step what to do, or to just throw you into it?)
By using the click to mark the smallest step and rewarding each little step toward your final goal you are breaking things down to make it easier for the animal to learn.
You are also able to "capture" behaviors. That is, mark and reward any good behavior your animal does all by them selves with out any prompting from you. (by capturing little behaviors I was able to teach Sasha to retrieve her toy and put it away in the toy basket without my fat butt ever leaving the couch
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The clicker is only used to teach new behaviors or to fine tune already learned behaviors, once the animal has the behavior the way you want it the clicker is no longer needed. The rewards are always at their highest during the learning period of training a new behavior, but once the animal has learn and practiced the behavior and understands it the rewards can be tapered off and mixed up.
Back to Sasha, after just a couple of months of using the chow chow who wouldn't sit was ready for her first obedience match (where by the way, she got for first qualifying score and won first place in her class beating out labs and shepherds) She is also much happier when it comes to training and gets way more excited about it. Even at the age of 7 she is still learning new things.
Side note: You do not have to use a clicker to "clicker train". People use a clicker merely because it is a quick consistent noise. You can substitute anything that makes a quick consistent noise, even a word (yes! Good! Zip! Beep!) as long as you use the same word in the same tone every time.
I've been training dogs for 9 years now. I've worked with obedience show dogs, regular family pets, and different aggresion cases and I have had very high success using clicker training in all cases. I do plan on using clicker training with my new horses.