Okay, let's start with what it's not: classical conditioning. Basically, classical conditioning is a form of learning where two stimuli become associated so strongly that the presence of only one of the stimuli will elicit the same response as if both were present. (If you need to know more, go read that node).
Operant conditioning is when you do one thing, another thing happens, and you therefore learn to do the first thing in order to cause the second thing to happen. This is stimulus-response connection. (classical conditioning is a stimulus-stimulus connection).
About the time Ivan Pavlov was working with his dogs, Edward L. Thorndike was messing about with cats; he put them into cages ('puzzle boxes') which had some trigger that needed to be pressed or pulled in order to open the door. The cats would struggle to escape, and by chance would hit or pull the right doodad; after a couple times the cats would learn to just hit the button or pull the string straight off. This response would be reinforced by repeated successes, and weakened by failure. This is called Thorndike's Law of Effect.
B.F. Skinner liked the general idea, and because of his work operant conditioning is largely associated with his name today. But Skinner was a Behaviorist, and so dropped all talk of intentions and desires on the part of the animal. He focused on the operant, which he defined as an observable, voluntary behavior that an organism emits to 'operate' (have an effect) on the environment. The Skinner-Box is a common apparatus used to experiment with operant conditioning.
Related ideas:
Reinforcement (which makes a response more common).
Reinforcers (the things that reinforce a response).
Conditioned reinforcers (A neutral reinforcer that works because it has been paired (by way of classical conditioning) with a primary reinforcer).
Reinforcement contingency (a consistent relationship of action/consequence).
Partial reinforcement (the desired response doesn't happen every time).
Partial reinforcement effect (extinction is delayed because the subject is used to partial reinforcement).
Intermittent reinforcement (the number of times reinforcement takes place has been predetermined, but reinforcement events are randomized).
Positive reinforcement: Giving a good thing (increases behavior).
Negative reinforcement: Taking away a bad thing (increases behavior).
Punishment
Positive punishment: Giving a bad thing (decreases behavior).
Negative punishment: Taking away a good thing (decreases behavior).
Extinction: The desired response doesn't happen (decreases behavior).
The Premack Principle: A more preferred activity can be used to reinforce a less preferred one.
Shaping: Approaching the desired behavior by reinforcing successive small steps approaching that behavior.
Autoshaping: Shaping without formal training by the experimenter.