The county is a local level of government below the state. Each of the 50 states is divided into counties (though in Alaska they are called boroughs and in Louisiana they are called parishes) that manage local governance. Each county has a county seat which is the center of county administration. Many counties are subdivided into self-governing municipalities.
Counties are viewed as agents of the states because they enforce state laws. For example, county sheriffs arrest violators of state laws, county clerks oversee state elections, and registers of deeds maintain state records such as property deeds and birth, death and marriage certificates.
"Counties trace their roots to the English of a thousand years ago. Serving a dual function, the shire acted as the administrative arm of the national government as well as the citizens' local government. The structural form of the shire was adopted along the eastern seaboard of North America by the colonists and adapted to suit the diverse economic and geographic needs of each of the colonies.
When our national government was formed, the framers of the Constitution did not provide for local governments. Rather, they left the matter to the states. Subsequently, early state constitutions generally conceptualized county government as an arm of the state." 
Systems of governing
Counties are usually governed by an elected board of supervisors, county commission, county council, or county legislature. In some counties, there is a county mayor or a county executive. The position of mayor is mostly ceremonial in some states, while in others, the mayor is more powerful than the commissioners or supervisors.
In many states, the board in charge of a county holds powers that transcend all three traditional branches of government. It has the legislative power to enact ordinances for the county; it has the executive power to oversee the executive operations of county government; and it has quasi-judicial power with regard to certain limited matters (like hearing appeals from the planning commission if one exists).
As for the day-to-day operations of the county government, they are sometimes overseen by a county manager or chief administrative officer who reports to the board, the mayor, or both.
In some states, the county technically has a plural executive in that several important officials are elected separately from the board of commissioners or supervisors (implying they cannot be fired by the board). This can create tension if such officials then disagree on how to best carry out their respective functions. 
- National Association of Counties Overview of County Government
- Wikipedia:County (United States) retrieved on 12/30/08