The head of the staff (CEO, Executive Director, etc.) of an organization has to answer to the Board of Directors. An organizational chart, for instance, would have a “Board of Directors” box over the top staff position. That person would then be over the department heads, and so on.
When listing Board members individually – for instance, in a newsletter masthead or an annual report -- it is customary to list the board officers (ie, President, Vice President, Treasurer, Secretary) first, then list the remaining board members in alphabetical order. Staff would be listed separately. If a person is both a Board member and staff person, as an Executive Director often is, several things could be done. Some organizations list this person at the end of the Board, others right below the Board officers, and still others just list them on staff (particularly if the Executive Director is a non-voting member of the board.)
Boards, however, do not have the same kind of “ranking” that staff do. An executive director can tell the rest of the staff what to do, within legal limits. The president of a board usually has certain administrative duties and powers over the rest of the board (the ability to call and adjourn meetings, for instance). However, board members real power rests in their vote, and no board member has more of a vote than another. In this sense, no board member is above the other. The exception to this is that in certain organizations, the president or chair of the board holds the tie-breaking vote. (In other cases, the executive director who usually does not vote would have this power). Of course, this flat hierarchy does not apply to an organization where voting power is based on the number of shares each individual has.