The click is a marker signal
Clicker training is a science-based system for teaching behavior with positive reinforcement. You use a marker signal (the sound of a toy clicker) to tell the animal (or person) when it's doing the action that will pay off. The system was first widely used by dolphin trainers who needed a way to teach behavior without using physical force.

"Clicker training" is a slang term used to describe a way of training animals that has become increasingly popular in the last decade because of its gentle methods. The scientific term for it is operant conditioning.

What is Operant Conditioning?
Operant conditioning is the way any animal (including the human kind) interacts with and learns from its environment. Simply put, an animal tends to repeat an action that has a positive consequence and tends not to repeat one that has a negative consequence. Trainers can take advantage of that natural tendency by providing positive reinforcement following an action that they want the animal to repeat. In order for the animal to connect the positive reinforcement to the behavior that he is doing, the reinforcement must happen AS the behavior is occurring, not afterwards. The actual reinforcement can't always be gotten to the animal at that precise instant, however.

Trainers needed to find another way of letting the animal know that he was doing the right thing, so they began using a conditioned reinforcer. A conditioned reinforcer is anything that wouldn't ordinarily be something the animal would work to get. A primary reinforcer, on the other hand, is something that the animal automatically finds reinforcing, such as food or water. When a conditioned reinforcer is paired with a primary reinforcer, they become of equal importance to the animal. Enter the clicker as a conditioned reinforcer.