A U.S. State is any one of the fifty states which is a member of the federation known as the United States of America. Sovereignty is divided between the individual states and the federal government. Under the United States Constitution, the federal government can legislate only on matters explicitly delegated to it by the Constitution, with the remaining governmental powers belonging to the states.
Originally, it was unclear whether a state could secede from the United States. It is the widely held perception that the American Civil War determined that it could not.
Unlike states the authority to rule those areas comes not from the people of those areas but from the Federal government, however in most cases Congress has granted a large amount of self-rule.
"Georgia" is the name of both a U.S. state and an independent country in Caucasia.
"New York" is all three: a state, a county in that state, and a city in that state.
"Washington" is a state, a city in the District of Columbia (and thus not part of any state), and a number of cities and counties in various states. The state Washington is the only one named after a U.S. President (or after a person born within the U.S., for that matter).
One state upon joining the United States retained the right to divide itself into up to four separate states. This was a condition of the statehood of Texas.
Only two states have state capitals named for the state (however, this is a very common practice with states and provinces in other countries, where the state or province is actually often named after the capital city): Oklahoma, with capital Oklahoma City, and Indiana, with capital Indianapolis (which means Indiana City).
