The names classical and operant conditioning refer to the two different ways dogs (and other creatures, even humans!) learn, the key difference between the two is that one is based on voluntary behaviour and one involuntary.
Classical conditioning is where a subject learns new behaviour based on an involuntary response to a stimulus, an example of this is a dog getting excited and jumping up at the sight of his leash, as the dog has been conditioned to associate the leash with going out on walks. When the dog was born, before he learned to associate the leash with a walk the leash was a neutral stimulus, meaning it had no effect on the dog. Over time as the dog learned he had to wear the leash attached to his collar before going out for a walk he has associated the leash with the enjoyment of going out for a walk, resulting in him getting excited whenever he sees his leash.
Operant conditioning is the learning method in which a new behaviour is learned through a voluntary action and a consequence of this action. An example of operant conditioning is a dog learning to bark on command. Initially the word ‘speak’ means nothing to a dog as he does not understand English, however by giving the dog a treat every time you say the word speak and he barks, he learns the association of the word speak and getting a treat. Both operant and classical conditioning needs to be reinforced regularly to be effective.