Is the United States a democracy or a republic?

This question is a favorite of online political discussions (truly the purest form of civil debate). When someone brings it up, their point is usually that the US is not a "true democracy" but rather a watered down form called a republic. Often, they will cite the fact that we do not directly vote on laws, but rather elect representatives who then vote on our behave. In the end, this is all hogwash as the United States is both a democracy and a republic. The question of democracy vs republic is analogous to asking if something is bright or yellow, i.e., both are independent qualities.

What is a republic?
The word republic derives from the Latin phrase res publica, which means "public matter". The term was invented by the Romans to describe their form of government between the period of rule by kings and rule by emperors (many emperors still used the term to describe their rule, but that doesn't make it true). A republic is a system of government where the head of state is both limited in power by laws, and elected by people. In practice, this means any state that isn't a monarchy or dictatorship is a republic.

What is a democracy?
The word democracy derives from the Greek word demokratía, or "rule of the people". In a democracy the people hold the power to rule. In practice, this means any state where the people are ultimately responsible for writing laws is a democracy.

Athenian democracy
It is true that these definitions I have given are different from the original classical definitions. This is, I think, the point many people are trying to make when they claim the US isn't a democracy. They are insisting that they would prefer a system of direct or pure democracy where people create and vote on laws directly without any representative body (e.g. Congress). This is all well and good, but to present the issue of one of either democracy or republic is misleading. A government can have both, neither, or either one. It is also probably worth noting that in the original Athenian democracy form, only 1 in 10 people could vote. If by virtue of being the original, this definition has more weight, then consider me quite glad we do not have a democracy.

Direct democracy
In modern usage, democracy has come to mean a system where people have a voice in the creation of laws. Either directly, as in direct democracy, or indirectly, as in representative democracy. There are examples of modern direct democracies, where (nearly) all the people of a state vote, with equal say, on laws.

Is direct democracy better than representative democracy?
One thing is certain, it is more popular. People like the idea of having power, and they like the idea of having control over themselves. I would support adding direct democracy elements (not the total removal of Congress). However, I am unswayed as to the superiority of direct democracy vs representative democracy. In the US many states have ballot initiatives, which are pretty close to direct democracy. Yet, those states don't differ very much from states without those initiatives. At the end of the day, our representatives do a pretty good job of accurately representing us. Or at least at representing what we care about enough to actually go vote about. Note, I am not saying they do a good job at representing our best interests (not by a long shot). Rather, they know what issues people care strongly enough to actually go vote about, i.e., big, flashy, popular issues that often have very little actual meaning.

People have a tendency to feel as if majority rule is the fairest system. However, it has many well known flaws. The core of the problem is that 51% of the people should not be able to impose their will upon 49%. On the other hand, the majority should have its way in many matters, but the more serious a matter is the more important it is to limit the majorities power. To use the extreme example: A group that is a majority shouldn't be allowed to vote to enslave a minority group. There is a quote that summarizes this well, "Democracy is two wolves and a lamb voting on what to have for lunch".