Classical conditioning is the behavioural process through which the subject will exhibit a behaviour in response to a conditioned stimulus; the conditioned stimulus having previously become associated with an unconditioned stimulus. The most famous example is that of Pavlov's dog. Pavlov rang a bell (neutral stimulus) before giving a dog food (unconditioned stimulus) - eventually, without giving the food, the dog would salivate at the sound of the bell (now a conditioned stimulus).
Operant conditioning is the process through which a behaviour is encouraged or discouraged, either through reinforcement or punishment. Operant conditioning can either be positive - meaning the addition of a stimulus - or negative - meaning the removal of one. The most famous example of operant conditioning using reinforcement are the experiments designed by Skinner. Skinner realised that rats will press a lever more if they receive food when they do - this is positive reinforcement, as the behaviour results in the addition of a reward. The same response occurred if it removed an unpleasant stimulus (such as a loud noise) - this is negative reinforcement, as the behaviour results in the removal of an unwanted stimulus. Whilst similar, operant conditioning tends to be associated with voluntary behaviours, whereas classical conditioning tends to be associated with involuntary behaviours (i.e. pressing a lever is a conscious choice, whereas salivating is not).