What does the FIDE president do? What is the role of the president of the International Chess Federation? What does he or she do, and what is he or she supposed to do? 

I have looked at sources such as Wikepedia
The FIDE President's role is defined in one of the FIDE Handbooks on the FIDE website. The relevant part is Chapter 09 - The President and Deputy President in the FIDE Statutes. 

There it says -


  9.1.  The President represents FIDE officially and can solely sign for FIDE.  All contracts signed by the President shall be subject to prior
  approval or subsequent ratification by the Presidential Board. Any
  contract that commits FIDE to an obligation or exposes FIDE to a
  potential liability of more than €100,000 must be submitted for
  ratification by the Executive Board or General Assembly.
  9.2.  The President presides over the meetings of the General Assembly, the Executive Board and the Presidential Board.
  9.3.  The President presents an annual report to the General Assembly.
  9.4.  The President’s decisions may be appealed to the General Assembly, Executive Board or Presidential Board.
  9.5.  If the President is unable to act or if he duly authorises, then he can be represented by the Deputy President who shall exercise the
  powers of the President. The Deputy President can thus represent FIDE
  officially and can solely sign for FIDE.
  9.6.  In certain cases the FIDE President can also delegate specific tasks to other persons; this includes authorising in writing other
  FIDE officials to sign contracts on his behalf.


Details of the General Assembly, the Executive Board and the Presidential Board are given in the FIDE Statutes.