I'm used to an Orthodox Union-type synagogue that has a board, a president, and a rabbi as an employee.
How does it work with a Lubavitch shul?
As I understand the Shul is generally owned by the Shliach (he sets up a (or uses an existing) non-profit for the legal aspects, but it's almost completely in his hands). This is done for several reasons:
There are no hard and fast rules. 770 has elections for Gabbaoyim (and Crown Heights in general has elections for Rabbonim). Some Shuls (in Crown Heights) have boards. In some, the old Rov bequeathed the position to a new one.