The most commonly known form of Democracy is the Representative Democracy, where the people elect representatives to act as a government. In theory, every form of a Democracy would give ultimate power to the people, in equal weight per person and in every matter. The concept of freedom and equality of citizens has changed drastically since the first form of Democracy in Ancient Greece; most forms today utilize some type of Representative government.
A Republic is superficially similar -- the people retain ultimate control over the government and there is no "king" to rule without limits -- but most Republics outside the United States are Sovereign or Unitary Nations, with the governmental structures basing their decisions entirely in the hold of power instead of legal or ethical reasons.
The form of a Republican Government in the United States is very similar to a Democratic Government, and uses aspects of each in an attempt to guarantee fairness to all citizens. Essentially, the United States Government is a Representative Democracy in which the people, through the Constitution, indirectly control their elected officials. The phrase Constitutional Republic is often used today to indicate a system of government in which all official, public positions are answerable to the people and are limited in their powers both by other branches of government (Checks and Balances) and the arbitrating law of the Constitution itself.
One interesting note comes from James Madison in his essay Federalist Number 10:
A pure democracy can admit no cure for the mischiefs of faction. A common passion or interest will be felt by a majority, and there is nothing to check the inducements to sacrifice the weaker party. Hence it is, that democracies have ever been found incompatible with personal security or the rights of property; and have, in general, been as short in their lives as they have been violent in their deaths.
The implication is that pure Democracy will always fall prey to majority rule, whether it is moral and ethical or not. Alterations of definition resulting from this statement are the root of the term "Republic" having a different meaning in the United States than elsewhere in the world.
The bottom line is that the United States government uses aspects of both traditional Democracy and traditional Republicanism. The Founders, in writing the Federalist Papers, weighed the pros and cons of each system and chose the best parts of each. Because of the changing nature of the language, we can refer to the United States either as a Democracy (representative) or as a Republic (constitutional) and be correct either way.