The board sets parameters of how the library will operate. There are few activities of the board team in which every segment of the board team isn't involved. For example, it is the board's job to establish an annual budget for the library. However, the budget is most often prepared by the director and recommended to the board for approval. So the director does have an important role in the budget process even though, strictly defined, it is a board responsibility.
Communication is the best prevention of confusion and conflict.
Board members and the director must feel free to discuss any issue and to express concern about who does what job. Board members never give up their bottom-line responsibility for the library even though they must delegate much of the work. The board awards bids for major purchases, but the director recommends which bid be accepted and actually makes the purchase. The board is responsible for making a long-range plan for the library, but the director will make recommendations and act as resource throughout the planning process. The board approves a budgeted amount for maintenance, but the director actually determines the maintenance activities that will be carried out.