Democracy is an overused term, as it is applied to revolutions and governments. We are a republic with democratic institutions. We elect people to make decisions for us because it would be too cumbersome and inefficient to have everyone participate in direct democracy. Everyone who is a citizen and 18 can vote. That's democratic. Everyone can go to school. That's democratic. And the Bill of Rights applies to everyone in the country.
The U.S. is not truly a democracy, we are a federal republic, specifically we are a democratic federal republic because our officials are elected by all the people. The term federal refers to the fact that we are in truth, a union of states with a federal government. This term may become more important in the near future if any of the states with secession legislation on their dockets actually tries to vote on this legislation. (See bills in New Hamshire, Arizona and several other states) The term republic comes from ancient Rome rather than Greece and a Republic is a modified type of Democracy. True Athenian Democracy does not work in a large state. It is unmanageable.
The definition of the word "democracy" is government by the people. "Demo" means people. "Cracy" is government. The United States is a limited democracy, since we elect leaders and representives to make laws for us, rather than voting on each issue ourselves, which would never work.
The Founding Fathers of our country did not use that word specifically because they based most of their ideas on English common law and the writings of people like John Locke. True democracy goes back to the time of ancient Greece, when Athens had close to a true democracy. Although of course, women and slaves could not vote, and they made up a large percentage of the population.
A democracy is different from governments such as an absolute monarchy, a dictatorship, or an oligarchy in that in a democracy the citizens have a clear voice in their government, either direct or indirect.