One thing I have noticed about conservatives is that they keep saying we are a republic and not a democracy. Actually, we are both. They aren't mutually exclusive by any means. A democracy is basically a system of government where the citizens either directly vote on legislation or they elect or appoint those who do so for them. We are of course the latter, a representative or indirect democracy. A republic is by its' most basic definition, is a non-monarchical state. Indeed, North Korea and China are republics because they aren't led by a monarch, though Kim Jung Il of North Korea is treated almost as a demi-god. The U.S. is a democratic republic or representative republic because while we are of course a non-monarchy, we elect people to legislate for us. We are also a constitutional republic because we have a constitution.
So, the question is, why do conservatives so strongly deny we are a democracy along with being a republic? Is it ignorance of basic political science? To a large extent, yes. One of the most ironic things in their opposition to the U.S also being a republic, is their strident opposition to judicial decisions that go against majority opinion. Court cases that struck down bans on interracial marriage, segregation or in some states bans on same sex marriage were met with anger by many on the right as going against the will of the majority. But wait, isn't that taking the position that we are a democracy and majority rules? Yes. One thing I have learned about many conservatives, is that they are full of opposing viewpoints that conflict. On one hand they are strident in the view we are not a democracy and in the next breath that a judicial court can't go against the will of the majority, even if the case involved the violation of civil rights of a disliked minority. Now, can those on the left sometimes do this, sure, but I think this cognitive dissonance is more common on the right.
For example, conservatives frequently condemn those on the left for opposing say, the Iraq war, and they often state those who are opposing the war are traitors or un-American. Many of those same people on the right though dissented (as their right) on wars they disagreed with, such as the Balkan conflicts. Sean Hannity and Rush Limbaugh are notorious for exactly this, and when pointed out to them (as I have personally) they get defensive.
I also think the term republic is used because of fear of those who are different from them. Maybe people who look and sound like President Obama, though they would strongly disagree with this assessment. They might even repress these views consciously. If we are a republic though and not a democracy, we can ignore the fact that African Americans in the South, who vote overwhelmingly for democrats in that region, can almost never get enough votes to elect a democrat (black or white) to the U.S. Senate in the South even though they make up 1/5 of Southerners.
Another speculation why conservatives like the term republic is that the Pledge of Allegiance refers to a republic. Also, republic is not Republicans and democracy is like democrats. Yes, trivial reasons such as that can motivate someone to have the opinions that they do.