Independence is the state of being free of the control of some other person, country or entity. Revolutions are all about obtaining independence, most famously perhaps the Revolutionary War in America, which led to America's freedom from Britain.
Independence comes from a nice medieval French word, depenre, meaning "to hang from," or "to hang down." The in at the beginning is Latin for "not," so the word originally meant "not hanging from," which is a neat description of what countries achieve by throwing off their colonizers. Teenagers also commonly seek independence from their parents — but perhaps not financial independence.