Independence is, literally, the freedom from dependence upon another person or body. It can be used to refer to a person living self-sufficiently, without the need for contributions from others, specifically one's parents ("living independently").
Politically, the term has come to mean freedom from control, whether by another country or by a despotic government. The United States celebrates its independence from Great Britain annually on July 4th, marking the signing of the Declaration of Independence at Independence Hall in Philadelphia.