Most Americans have more daily contact with their state and local governments than with the federal government. Police departments, libraries, and schools—not to mention driver's licenses and parking tickets—usually fall under the oversight of state and local governments. Each state has its own written constitution, which are more elaborate than their federal counterpart.

State Government

Under the Tenth Amendment to the U.S. Constitution, all powers not granted to the federal government are reserved for the states and the people. State governments are modeled after the federal government and consist of three branches:

Executive: headed by a governor who is directly elected by the people. In most states, the other leaders are also directly elected, including the lieutenant governor, the attorney general, the secretary of state, and auditors and commissioners.

Legislative: All 50 states have legislatures made up of elected representatives, who consider matters brought forth by the governor or introduced by its members to create legislation that becomes law. The legislature also approves a state's budget and initiates tax legislation and articles of impeachment.

Judicial: State judicial branches are usually led by the state supreme court, which hears appeals from lower-level state courts. Court structures and judicial appointments and elections are determined either by legislation or the state constitution. The Supreme Court focuses on correcting errors made in lower. Inconsistencies with the U.S. Constitution, may be appealed directly to the United States Supreme Court.

Local Government Local governments generally include two tiers: counties, also known as boroughs in Alaska and parishes in Louisiana, and municipalities, or cities/towns.

Municipalities generally take responsibility for parks and recreation services, police and fire departments, housing services, emergency medical services, municipal courts, transportation services (including public transportation), and public works (streets, sewers, snow removal, signage, and so forth).

Whereas the federal government and state governments share power in countless ways, a local government must be granted power by the state. In general, mayors, city councils, and other governing bodies are directly elected by the people.

Source: State and Local Government
The White House, usa.gov