@Baldimo,
There is direct democracy, which is when things are voted on by everyone. There is representative democracy, where we elect the people who vote on things for us. We live in a country of 320 Million people. A direct democracy is not possible.
A Republic is a country without a monarch system of government. So we live in a republic and a representative democracy. We covered this stuff in fourth grade and each year thereafter. Somehow conservatives have gotten hold of this "we don't live in a democracy, we live in a republic" thing and are angrily waving it around like they have some vital point to make. They don't.