First of all, technically, isn't the United States of America a mix of both worlds, democracy AND republic? In school, my teacher said that this was partially true, because we don't leave EVERYTHING up to the people. The government does a lot of work, too. So, WHAT IS IT exactly?
America is a democratic republic - we democratically elect representatives to run our government. If we were a pure democracy, every citizen would have to vote on every issue - it would be a logistical nightmare, to say the least.