We are a form of democracy, but not a pure democracy. Pure democracies don't work.
My copy of the Constitiution says that Congress must guarrantee to the states a Republican form of government.
"Democracies have ever been spectacles of turbulence and contention; have ever been found incompatible with personal security or the rights of property and in general have been as short in their lives as as they have been violent in their deaths." Madison in Federalist Papers #10
For further information on Democracies try The Sequence of Democracy by Alexander Tyler written 1770. It's amazing how close it comes to what we have today.