A red herring in a comment made by Shadowhelm a while back, started me thinking about the different ways in which we can define our government. He said our government was not a democracy but a republic. My initial response was to think, "damn, everybody calls themselves a republic. What does that tell you?"
So it has been in the back of my brain to look up the definitions of 'republic' and 'democracy' as well as the different types that could apply to the United States of America. Since I don't feel like doing actual work this morning, I decided I'd look them up (using wikipedia and a couple of online dictionaries).
The word "republic" comes directly the French and its most basic meaning is a state in which supreme power rests in the people. There are some modification to this definition in modern usage making the best definition I've found to be: a state in which the supreme power rests in the body of citizens entitled to vote and is exercised by representatives chosen directly or indirectly by them. Additionally, the head of state can not be a monarch, although the United Kingdom can, as a constitutional monarchy, be considered one type of republic. As you can already see, definitions of political systems can be complex and confusing.
The word "democracy" is derived from the Greek meaning rule of the common people. Key parts of the modern definitions include majority rule, free elections, and the power of the people. Most modern democracies are representational, where the power of the people is exercised by their representatives and not the individuals personally. Most people think of Athens as the first democracy but the Athenian government would probably not fit the description of a democracy by those same people. The number of "citizens" was relatively small (between 10-15% of all residents) and not all had the same vote. Additionally, Athens would be a direct democracy (if one at all) and most modern democracies are representational.
From my (not vast) reading the USA is a federal and constitutional republic. We can also be said to have a mixed government/constitution. Some other would say we are a capitalist republic (Marx's "dictatorship of the bourgeoisie"), but those would be people who aren't fond of our economic system. In terms of democracies, we are a representative, or more specifically a liberal democracy. (Please note that almost all of the world's government's refer to them selves as republics and/or democracies, whether anyone else would define them as one or not.)
That brings an end to my lesson on types of governments. Regardless of how one names the government, a representational government requires an educated electorate or it will become an oligarchy, ruled by the those already in power and their hand picked successors.