The traditional Pavlovian classical conditioning teaches us that a response can be achieved by association. In humans, it means that a person can achieve a behavior, or respond to a situation in a specific way, as long as there are stimuli that trigger such a reaction.

For example, an associative situation would be when a person hears the sound of a can opening, and the salivary glands activate associating the sound of the opening can to a nice, cold drink. Contrastingly, if someone detests soda and hears a can opening, feelings of disgust or dislike might be produced. All this is what association is about.

The Pavlovian classical conditioning theory also states that when a stimulus has not been conditioned to trigger a reaction, the stimulus is known as an "unconditioned stimulus", or a UCS. An example of a UCS would be when dogs casually smell food and start to salivate. The smell of food is an everyday incident in the lives of domesticated dogs. Therefore, the reactions of the dogs are natural and predictable.

When the stimulus has been prepared  to draw a specific reaction from an individual, then that stimulus becomes known as a "conditioned stimulus" , or CS.  Similarly, the reaction that the conditioned stimulus produces is known as a "conditioned response", or CR, because it is part of an experimental process.

An example of a CS would be using an object to cause pain or fear. If the object has been conditioned to instill feelings of fear in an individual, then the object takes a new meaning for the person, creating a reaction of anxiety or fear when the object is shown.

In a different note, classical conditioning also shows us that dissociation can be used to break the associations that may trigger negative responses in humans. For instance, dissociative therapy may help some patients control common phobias by removing the negative stimulus away from an object that once caused fear.

Therefore, classical conditioning is a great way to understand and predict human behavior.

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.