What are the duties of a board of directors of a homeowners association?

The general and specific duties of a board are to carry out the responsibilities of the association. Those duties should be enumerated in the association's bylaws (sometimes they will be found in the articles or CC&Rs) and will usually include things like financial planning and management including overseeing creation of financial documents such as budgets and reserve plans, maintenance of the property, and enforcement of governing documents. It will also usually include managing the day-to-day operations of the association, but often much of these responsibilities are delegated to the association manager.