In behaviorism, extinction is what happens when a psychologist makes a subject stop exhibiting a behavior in response to a given stimulus. In the famous example of Pavlov's dogs, extinction would be if the researcher made the dogs stop salivating when they heard the bell. When their conditioned response to the bell was gone, extinction would have occurred.
Test subjects can be made to respond in given ways to given stimuli. In the case of the dogs, they came to associate the bell with food and so they would salivate when they heard the bell. This is a conditioned response. To erase the response (to cause extinction) the psychologist must keep exposing the subject to the stimulus (the bell) but stop reinforcing the conditioned response (stop giving the food). After a while, the subject stops connecting the stimulus with the response and extinction has occurred.