If you live in a republic, then you live in a country with a leader freely and democratically elected by the people, as opposed to a dictatorship or monarchy.
Ancient Rome, before it started declining and falling all over the place, was a republic, and so is the United States. Republics have what are termed "republican" forms of government — not to be confused with Republican-with-a-big-R, as in the American political party. If you live in a "Banana Republic" you probably live in a tiny Caribbean island run by a dictator. Or you're spending too much time at the mall.