Once the mage has been successful with the first step of communication, the elemental will make its way to his the wizard.  In this example, we see that that the wizard was able to communicate with a Celebane.  Although the Celebane is a frightful looking elemental, it is very gentle.  In the image above, she has arrived with her child.  

The elemental (almost all of them) are generally very skittish and will look around to see who has called her.  This is the hardest part of the communication process... first impressions.  The wizard has to gain the trust of the elemental. At this time, he'll refer to his Oracle cards to determine what the elemental wants.  

Once the elemental receives the gift, she has the option of staying or leaving.  Like many people, they have a "sixth sense" about whether to trust the individual or not.  At this point, further communication is up to her.  If she is not interested, the beginning mage will need to rest before attempting another communication.  As the wizard gains more experience, the days of rest shorten.