While this is very interesting from the academia point of view, how is it useful in dog training? To understand how the classical conditioned click sound is helpful to train your dog, you must be familiar with the concept of operant conditioning, so that is what you are going to learn now.
Operant conditioning is a process discovered by the American psychologist Burrhus Frederic Skinner. According to operant conditioning, those behaviors that have pleasant consequences tend to be repeated. Thus, their frequencies increase and the behaviors are strengthened.
In a similar way, those behaviors that have unpleasant consequences tend to be avoided. Therefore, their frequencies decrease and the behaviors tend to disappear.
Thus, a behavior is reinforced (strengthened) if it has pleasant consequences. Pleasant consequences are called positive reinforcers, since their occurrence strengthens the behavior of your dog.
Positive reinforcers are commonly known as rewards or treats. However, those are not right terms and it is better to get familiarized with the proper term, positive reinforcer, which is widely used in clicker dog training literature.
On the other hand, a behavior is also reinforced (strengthened) if it removes an unpleasant consequence. In this case, unpleasant consequences are called negative reinforcers, because their removal strengthens the behavior of your dog.
So, according to operant conditioning, you just need to reinforce a new behavior of your dog to make it stronger and more frequent. Positive reinforcers must appear at the same time or immediately after your dog does the behavior you are teaching to him. Negative reinforcers must disappear at the same time or immediately after your dog does the behavior you are teaching to him. Otherwise, the reinforcers have no effect.
It is not easy to achieve an optimum timing in practice, and that is the reason why many trainers use a conditioned reinforcer. A conditioned reinforcer is a signal that allows you to mark the moment when your dog performs the desired behavior. This mark tells your dog which behavior is going to be reinforced. The conditioned reinforcer used in clicker dog training is the click sound.
As you may imagine, a click (or any other signal) is meaningless to your dog if it has not been taught before. That's why the first step in dog training is the creation of a conditioned reinforcer through classical conditioning.
There are some typical conditioned reinforcers in dog training. One of the most popular these days is the click sound of a clicker. Nevertheless, it is not necessary to use a clicker. Some trainers prefer to use words instead of a clicker.
If you decide to use a verbal marker instead of a clicker, avoid long phrases like "Good dog". Use a short and sharp word that lasts just a fraction of a second, like "Ok". Thus, your dog won't be confused if he displays several behaviors in a short time.