A demotic saying or expression is casual, colloquial, and used by the masses. Some forms of the Greek and Egyptian languages are also called demotic, which will be relevant to you when you get your PhD in Classics.
Demotic comes from the Greek word demotikos, meaning “of or for the common people” or “in common use.” Members of the aristocracy don’t typically use demotic idioms, but it is often the elite who will point out that something is demotic. Of course, in a classless society, everything ought to be demotic, therefore making it obsolete to designate sayings as demotic. So far, however, demotic is still a relevant term.