The board of directors is the group of people that runs/owns a football club. It is made up of a chairman and a number of other people such as a managing director. On FM, most large clubs have a board of directors, whereas most smaller clubs just have a chairman.
If a club hasn't got a chairman or any directors set up, news items will refer to the club's "board," by default, without mentioning the names of any staff.
In Football Manager
Directors were included for the first time in Championship Manager 3 and have been a feature of the game ever since. Up until Football Manager 2007, directors (and other club officials) could only be changed using a data editor but from FM2007 onwards board takeovers can occur during the game.