DreamsMaster: A president generally represents someone with power and control. To shake hands with the president in a dream could mean that you’re getting in touch with your own inner power and have a sense of control over your life right now. The fact that the two of you are engaged in lively conversation suggests that this sense of power and control are manifesting in your life in the areas of creativity and communication.
Is it clear in the dream which president, specifically, you are shaking hands with? If so, is he the president of the country you currently reside in, or does he preside over a different country? Is this president someone whose political views you generally agree with, or not? These additional pieces of information can add further clarity to the dream’s meaning. For example, if it’s a president you associate with a specific political position that you’ve protested in the past, shaking hands with him in your dream may mean your position or attitude about that subject has become more aligned with his.
When you awoke from the dream, did you have any recollection of the conversation itself? Also, was this “lively conversation” friendly in nature, or was there an edge of conflict or hostility? The nature and tone of the dialog could shed additional light on the dream’s meaning as well.