"Here is an example of how the founding fathers talked about democracy.  It is from Patrick Henry at the Virginia convention to ratify the Constitution:

'A trifling minority may reject the most salutary amendments. Is this an easy mode of securing the public liberty? It is, Sir, a most fearful situation, when the most contemptible minority can prevent the alteration of the most oppressive Government; for it may in many respects prove to be such: Is this the spirit of republicanism? What, Sir, is the genius of democracy? Let me read that clause of the Bill of Rights of Virginia, which relates to this: 3d cl. "That Government is or ought to be instituted for the common benefit, protection, and security of the people, nation, or community: Of all the various modes and forms of Government, that is best which is capable of producing the greatest degree of happiness and safety, and is most effectually secured against the danger of mal-administration, and that whenever any Government shall be found inadequate, or contrary to these purposes, a majority of the community hath, an undubitable, unalienable and indefeasible right to reform, alter, or abolish it, in such manner as shall be judged most conducive to the public weal." This, Sir, is the language of democracy; that a majority of the community have a right to alter their Government when found to be oppressive: But how different is the genius of your new Constitution from this? How different from the sentiments of freemen, that a contemptible minority can prevent the good of the majority? If then Gentlemen standing on this ground, are come to that point, that they are willing to bind themselves and their posterity to be oppressed, I am amazed and inexpressibly astonished.'

To the founders, democracy had two meanings: a) all forms of self-government and b) the process by which the people directly made law.  An example of the second definition is the New England town meeting where the people directly made law.  Today, in most states, people vote on certain ballot initiatives.  This was often called "pure democracy."

The national government is a republic not a pure democracy.  However, the founders would refer to the national government as a democracy, but this would be in the first sense--as being a form of self-government.

Democracy is like the term "automobile."  Republic is like "Chevrolet."  Republic is a particular kind of democracy.  Our Constitution created a republic."