The Constitution of the United States acts as the rule of law and is the foundation of the American Government system. It outlines three branches of government -- Legislative, Executive, and Judicial -- intended to act as balances to each other and prevent any one part taking control. By allowing all parts of government a say in law and law-making, but ultimately deferring election of representatives to The People, the Constitution forms a Representative Democracy, or Constitutional Republic, instead of a monarchy or socialism.
The words "Democracy" and "Republic" have different meanings in the U.S. than in other parts of the world. Democracy refers to any of three types: Consensus, Direct, and Representative. Historically, Democracies have fallen because specific parts -- whether they be classes, rulers, or courts -- have become too powerful and were able to control the other parts. A Republic, on the other hand, is a system where the people have ultimate control over the government. Many republics around the world are sovereign, socialist, or dictatorial.
In the U.S., the terms Democracy and Republic are almost interchangeable when referring to a system of government. James Madison, who wrote most of the Constitution and Federalist Papers, specifically defined Democracy in Fed #10:
...a pure Democracy...can admit of no cure for the mischiefs of faction. A common passion or interest will, in almost every case, be felt by a majority of the whole...and there is nothing to check the inducements to sacrifice the weaker party...
Here, you can see his argument that a Democracy will always eventually bow to the wishes of majority rule, and therefore a minority will always be unrepresented.
Madison goes on to say:
The two great points of difference, between a Democracy and a Republic, are, first, the delegation of the Government, in the latter, to a small number of citizens elected by the rest: Secondly, the greater number of citizens, and greater sphere of country, over which the latter may be extended.
The important thing here is that Madison's Republic is very similar to a Representative Democracy, but with more emphasis placed on the actual representing of the people by the government.
Meanwhile, while drafting the Constitution, Madison made a similar statement in Article 4 Section 4:
The United States shall guarantee to every State in this Union a Republican Form of Government, and shall protect each of them against Invasion; and on Application of the Legislature, or of the Executive (when the Legislature cannot be convened) against domestic Violence.
Again, the difference between Democracy and Republic is small, but the line between the two branches mentioned (Executive and Legislative) is made clear; there can be no one branch controlling the others.
Ultimately, the Constitution was intended to establish a Constitutional Republic, which as defined in the U.S. is very similar to a Representative Democracy. It was never intended to establish Pure Democracy (as defined by Madison) or to allow anarchy through majority or mob rule. In the sense that it is a government created by the people, and subordinate to same, it is a Democracy, but since there are clearly defined parts and clearly stated powers, as well as the citizen's freedom to choose their representatives, it is not Classical or Pure Democracy.