I asked this question Can check (choke) chains still be used to successfully train dogs? and wanted to address specifically the use of a prong collar.
(image courtesy of Wikipedia)
Is it harmful to use a prong collar and if so, why?
First we could say that yes, this is clearly harmful, and that's the whole point of it !
Then we should ask ourselves "do I want to harm my dog" ? This depends on your own point of view.
But then, more importantly: do we need to harm a dog to train it ?
Many methods that were used in the past, and that are still used in many places are based on outdated concepts and training methods: strongly jerking the leash, using choke collars, etc.
Nowadays most successful trainers use so-called "positive reinforcement methods": from dolphins, chickens, horses, humans (google "tag teaching"), to dogs. These methods are based on real and up-to-date scientific results (applied psychology meeting ethology) and are constantly evolving and being refined. As an introduction to this I suggest to read this.
Answering the question "do we need to use chain collars to be successful" also avoids the endless debate "yes it hurts - no, it's not too bad, etc.". People arguing to use these tools are constantly getting closer to abusing their pets.