Under a board's legislative powers, the supervisors have the ability to pass and repeal laws, generally called ordinances, as in cities. Depending on the state, and the subject matter of the law, these laws may apply to the entire county or only to unincorporated areas not under jurisdiction by a city. The board is also responsible for approving the county budget. County governments frequently collect state taxes and, in some states, they or their citizens are allowed to set certain tax rates. However, because they are ultimately organs of state law and policy, it is unusual for counties to have power to establish their own taxes - they, or their citizens, merely adjust the rates of tax measures created by the state government. Cities, by countrast, typically have power to create their own taxes. Detroit, for example, has its own income tax, and New York has its own sales tax, on top of the New York State sales tax.
In some states, including Michigan, and in some New York counties until recently, county governing boards were composed of township (Michigan) or town (New York) 'supervisors'. These were the chief elected officials of each civil township (called 'town' in New York) in a county - hence, the term "Board of Supervisors", because they were originally composed of the various town/township supervisors from across the county. This system gave every township one vote on the county board regardless of its population, resulting in (usually rural) townships with few people having influence in decision-making that was disproportionate to their populations. A township with fewer than 1,000 people had the same clout as a city with more than 1,000,000 people.
Both Michigan and New York changed how they elected county boards by dividing counties into single member districts, drawn so that each district has more or less the same sized population. In Michigan, the new board model was implemented in 1968. The name 'Board of Supervisors' was changed to 'Board of Commissioners' in 1970, however, to avoid confusion with township government (where the term 'Supervisor' was still used). In New York, the new boards were called "county legislatures" (and their members, "county legislators"), but not every county has adopted this system. Those which retained the old boards of supervisors after the 1960s assigned each member a proportional vote based on the population represented.
Under a board's executive powers, the board controls county departments. Generally this is done under the aegis of a chief administrative officer or county executive. The power of the CAO or county executive to act independently depends on the composition of the board. Generally, like most city managers, the CAO or county executive has authority over the day to day operations of the county's departments. Many boards independently appoint department heads, while other boards may delegate that authority to the CAO or chief executive. Some department heads, like the sheriff or district attorney, may be elected separately by the electorate; however, the board still exerts some power over these department's budget. Despite the presence of a CAO or chief executive, it is not uncommon for an individual supervisor to meddle in the affairs of individual departments, like setting priorities for projects in one's district or independently requesting investigations of problems in a department. In some counties, the county executive is elected.
In the City and County of San Francisco, which, as the legal name indicates, has a consolidated city-county government, the Board of Supervisors does double duty as a county board of control and a city council, while the mayor, likewise, is simultaneously city head of state and county executive. Or to put in cross-comparative terms, San Francisco is the only city in California with a county Board of Supervisors instead of a city council; and it is the only county in California with a mayor instead of a county executive.
Under a board's quasi-judicial capacity, the board is the final arbiter of decisions made by commissions underneath a board. This generally involves land use planning issues.