Behaviorism is a learning theory that emphasizes routines, rewards, and punishments, and so has fallen out of favor in many modern classrooms. However, at its most basic level, the main objective of behaviorism is to study observable, measurable behavior. It's hard not to see how this approach would be at least partially beneficial to educators.
Ivan Pavlov was the first scientist to discover the principle of "classical conditioning." In a now famous experiment, he studied his dogs' reactions to food. As expected, when presented with food, they would salivate. This is a totally natural stimulus-response reaction. Nothing new here.
Pavlov went on to then ring a bell every time he presented food. The bell is what he called a "neutral stimulus" because by itself, it should not cause the dogs to salivate. However, as time passed, Pavlov took the food away, and simply rang the bell, the "neutral stimulus," and found the dogs salivated at the sound, even in the absence of food. Pavlov, through association and positive reinforcement, had created a new "stiumlus-response" pattern in his pets.
J. B. Watson was the psychologist who came up with the term "behaviorism." He thought that psychology should be based solely on objective, observable behaviors, which was divergent from prominent psychologists at the time, such as Freud, who based many of his theories on the subjective, subconscious mind.
Finally, it was B.F. Skinner who brought behaviorism to the mainstream. According to his theories humans, just like animals, respond to stimuli. Therefore, if we want to understand humans, we must observe a behavior, then look back at what stimuli created that behavior. Furthermore, observed behaviors could be reinforced with positive or negative reinforcement.
There are several pros and cons to behaviorism as a learning theory. When an educational goal is concrete and objective, such memorizing vocabulary or meth facts, then this simple approach works well. The student is given a goal, and then when acting upon that goal, the teacher can give positive or negative reinforcement until the goal is met.
However, learning doesn't always follow such a clear cut path, and certain assessments will be subjective, such as portfolio or project-based learning. In these types of situations, behaviorism as a working system is far too simplistic. That's not to say that we don't try to still use behaviorist principles when designing learning goals, rubrics, and even progress reports.
Overall, behaviorism is a useful learning theory will applied to learning tasks that are straightforward, objective, and easily measurable. Therefore it is best applied on activities at the bottom of Bloom's Taxonomy.