In behavioural terms, extinction is the lack of any consequence following a behaviour. When a behaviour is inconsequential (i.e., producing neither favourable nor unfavourable consequences) it will occur with less frequency. When a previously reinforced behaviour is no longer reinforced with either positive or negative reinforcement, it leads to a decline in that behaviour.
In other words, if your dog has been reinforced for a behaviour, and that reinforcement is removed, the behaviour will eventually decline or extinguish. Reinforcement does not necessarily always come from you but often comes from the environment. For example, if your dog jumps on the kitchen counter top and finds food there, then jumping on the counter top behaviour will happen again. Every time this behaviour leads to reinforcement it will become stronger. You did not play an active part in this scenario. “Catching the dog in the act” and punishing it will only result in the dog being more careful to indulge in this “counter surfing” when you (the punisher) are not present. What will work in this case is to remove the reinforcement (food on the counter). If the counter NEVER provides food for the counter surfer, the behaviour will extinguish.
Bear in mind, however, if a behaviour is either self-rewarding or rewarded by the environment, extinction will fail unless the trainer can consistently offer a better reward. Barking is frequently a self-rewarded behaviour. Waiting for a dog barking at passers-by to simply get bored from “lack of reinforcement” is an exercise in futility. When dealing with self-rewarding or environmentally-rewarded behaviours, a combination of management, positive reinforcement, and negative punishment is an alternative solution.
Just before behaviour extinguishes it undergoes an “extinction burst”. This means that the behaviour will become more and more intense just before it goes away. If your dog barks incessantly to be let in and you decide to use extinction as a way of stopping the behaviour, be prepared for it to increase in intensity, duration and volume just before it extinguishes. If you open the door at this point, you have reinforced the behaviour at its worst and this is what you will get next time. The only way to deal with this is to carry on ignoring it. Fit ear plugs, retreat to the furthest part of the house, and warn your neighbours.
Savvy trainers use this extinction burst as a tool when raising criteria in certain behaviours. For example, say we are working on latency (speed) of the sit. We ignore all the slow sits and only reward the fastest sits. What will happen is that in response to the absence of reinforcement the sits will become faster as the dog’s sitting behaviour starts to undergo an extinction burst. Because these more intense sits are the only ones that are being reinforced, the slower sits will extinguish and the better ones will increase.