According to contemporary psychology, what is actually learned druing classical conditioning?
Ivan Pavlov defined Classical Conditioning in 1927. Pavlov used dogs as subjects and exposed them to a neutral stimulus (in this case a ringing bell) followed by an unconditional stimulus (meat powder). The result was the “conditioning” of the dog to salivate when the bell rang.
Thus, in classical conditioning, the subject learns that the neutral stimulus is connected to the unconditional stimulus. As a result, the subject reacts to the neutral stimulus the same way he normally would with the unconditional stimulus, a so-called "conditioned response". Prior to the conditioning, the subject would not have learned the association between the bell and the meat powder. He would not have learned to salivate when the bell rang.
So the answer to your question is that conditioning causes the subject to learn an association between a neutral stimulus and something else, when before such conditioning he would have been unaware of the connection.