The United States is a Republic - or at least we were created as a de jure Republic. Now, we're actually more like a de facto legislative democracy due to corruptions like the 16th and 17th amendments and court decisions that allow government to basically do whatever it wants to do.
Just to show the real differences between a Republic and a Democracy, here is one of the most thorough definitions of those terms I've seen. It is from the 1928 U.S. Army training manual:
"Democracy: A government of the masses. Authority derived through mass meeting or any other form of ‘direct’ expression. Results in mobocracy. Attitude toward property is communistic – negating property rights. Attitude toward law is that the will of the majority shall regulate, whether it be based upon deliberation or governed by passion, prejudice, and impulse, without restraint or regard to consequences. Results in demagogism, license, agitation, discontent, anarchy."
"Republic: Authority is derived through the election by the people of public officials best fitted to represent them. Attitude toward property is respect for laws and individual rights, and a sensible economic procedure. Attitude toward law is the administration of justice in accord with fixed principles and established evidence, with a strict regard to consequences. Avoids the dangerous extreme of either tyranny or mobocracy. Results in statesmanship, liberty, reason, justice, contentment, and progress."
The following is an editorial note by Harry Atwood in the manual: "A republic is a form of government under a constitution which provides for the election of (1) an executive and (2) a legislative body, who working together in a representative capacity, have all the power of appointment, all power of legislation, all power to raise revenue and appropriate expenditures, and are required to create (3) a judiciary to pass upon the justice and legality of their governmental acts and to recognize (4) certain inherent individual rights.
Take away any one or more of those four elements and you are drifting into autocracy. Add one or more to those four elements and you are drifting into democracy.
Autocracy declares the divine right of kings; its authority can not be questioned; its powers are arbitrarily or unjustly administered.
Democracy is the ‘direct’ rule of the people and has been repeatedly tried without success.
Our Constitutional fathers, familiar with the strength and weakness of both autocracy and democracy, with fixed principles definitely in mind, defined a representative republican form of government. They made a very marked distinction between a republic and a democracy, and said repeatedly and emphatically that they had founded a republic.”