A republic is a kind of democracy; it is a form of representative democracy, as opposed to a direct democracy. "Democracy. n. 1. government by the people; a form of government in which the supreme power is vested in the people and exercised directly by them or by their elected agents under a free electoral system." http://dictionary.reference.com/browse/democracy
A republic is not necessarily a democracy, and the two terms are not strictly interchangable. A republic may take many forms, including but not limited to democratic, mercantile, classical, and religious republics.
Constitutionally the USA is a representative democracy defined as republican in modern political argot. If you have the time a reading of Federalist No. 10 is quite enlightening (although a little tough).
