Answer:
The United States government is a Republic, or more specifically, a federal constitutional republic.
Answer
The following is what the founding fathers wrote in the US Constitution and believed:
Article IV 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.
Thomas Jefferson said, "A democracy is nothing more than mob rule, where fifty-one percent of the people may take away the rights of the other forty-nine."
Thomas Jefferson, March 11, 1790: "The republican is the only form of government which is not eternally at open or secret war with the rights of mankind."
Alexander Hamilton, in debate, said: "Real liberty is neither found in despotism or the extremes of democracy, but in moderate government."
Alexander Hamilton, in Senate: "It has been observed that a pure democracy, if it were practicable, would be the most perfect government. Experience has proved that no position is more false than this. The ancient democracies, in which the people themselves deliberated, never possessed one feature of good government. Their very character was tyranny: their figure deformity."
John Adams, in a letter to John Taylor, wrote: "Remember, democracy never lasts long. It soon wastes, exhausts and murders itself. There never was a democracy yet that did not commit suicide."
James Madison said: "...democracies have ever been spectacles of turbulence and contention; 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."
Answer
Reduced to its essence the United States is a federal republic, a federation of states with a republican form of government, i.e., a representative democracy, where the citizens elect their representatives for fixed terms of office.
The Constitution of the United States, Article IV, ยง 4 provides: "The United States shall guarantee to every State in this Union a Republican Form of Government...." The requirement that the Constitution contain a provision guaranteeing a republican form of government seems to have been driven by Madison. Thus, it is appropriate to turn to his definition of a republican form of government in The Federalist Papers, no. 39, 250--53 (January 16, 1788). There, he writes that "we may define a republic to be, or at least may bestow that name on, a government which derives all its powers directly or indirectly from the great body of the people; and is administered by persons holding their offices during pleasure, for a limited period, or during good behaviour."
A familiar contrast is often made between a republican and a democratic form of government based on the latter's being a direct wielding of power by the people with unrestricted majority rule, and the former's being a government by representatives elected by the people with restrictions on the power of the majority. It should be noted that the requirement of restrictions on the power of the majority are not contained in Madison's definition. The restrictions on the power of the majority are, instead, contained in the Bill of Rights. Thus, whichever definition one adopts, the United Stated has a republican form of government.