By far the most common form for a nonprofit organization includes a volunteer board of directors and a staff made up of employees and (often) other volunteers.
Since nearly every nonprofit organization has a board with several members and there are literally hundreds of thousands of nonprofits, there are a lot of people with an interest in how boards should operate and there are many resources available to help develop and maintain effective boards overseeing effective organizations. This page provides some very general background. At the end, there are some annotated links to guide further exploration.
The board of directors has overall responsibility for the conduct of the organization. There are some broadly accepted standards about the way boards are supposed to work; many state laws governing nonprofit organizations include further requirements or standards that vary from state to state.
The first principle is that the board operates solely as a group. Boards take official action at board meetings; except at board meetings, the individuals who serve on a board have no special powers unless they are also officers or the board has voted to give ("delegate") responsibility for some task or area of work.
The second principle is that board members owe the organization careful attention in the conduct of the board's work (the "duty of care") and must always act in the organization's best interest (the "duty of loyalty"). When a board member has a conflict of interest—in other words, when a board member might receive some personal benefit as a result of a pending decision—he or she is required to explain the circumstances and avoid taking any part in the decision (stepping back in this way is often called "recusal"). IRS rules impose penalties on people (directors and others who work with nonprofits) who receive undue benefits — the rules governing this subject deal with "Intermediate Sanctions." For more about Intermediate Sanctions (including links to the IRS materials) see http://en.wikipedia.org/wiki/Intermediate_sanctions.
More generally, boards are responsible for the overall health and effectiveness of the organization. Observers have developed a short list of further "key duties" of the board; for example, here is the list of "Ten Basic Responsibilities of Nonprofit Boards" from BoardSource:
The responsibilities, powers and general operating procedures for the board of each nonprofit organizations will be described in its "Charter Documents" and in other rules, directives and manuals. Another page offers some examples of charter documents.
The board, and its members, can be held to account by government officials (the state Attorney General and the IRS, among others) and can be sued for failing to meet their responsibilities with care or with loyalty. Boards often purchase Directors and Officers ("D&O") Insurance to reduce the risk of personal loss for the individual board members if that should happen. Insurance can only protect against losses when the board or board member acted "in good faith," which explains to some extent why paying attention to potential conflicts of interest is a good idea. Another page has some examples of conflict of interest policies.
There are many theories and models describing how a board can best serve the community while overseeing a nonprofit and its work. Ultimately, each board is responsible for devising its own operating style and for evaluating its own, and the organization's, effectiveness.
Boards have to develop procedures for their own work and to govern their relations with staff, volunteers, clients, stakeholders, and the community at large. "Rules of order" (often based on the parliamentary procedures codified in "Roberts Rules of Order") are necessary to govern board meetings and other official functions.
Unfortunately, some boards fall into a pattern of internal conflict or conflicts between board and staff. There is no universal approach that can guarantee this won't happen, and no one-size-fits-all remedy that can be applied if it does. When the organization's services matter to the community and the conflict is impairing delivery and effectiveness, everyone concerned will want to focus on finding a solution as quickly as possible. Sometimes this will require painful adjustments in familiar patterns and routines.