Our Constitution created a limited representative republic. A republic is different from a democracy. In a democracy, the majority can directly make laws, while in a republic, elected representatives make laws. Basically, in a pure democracy, the majority has unlimited power, whereas in a republic, a written constitution limits the majority and provides safeguards for the individual and minorities.
In the United States, we actually have both systems. There is no way for Americans to directly enact legislation at the national level, but half of the states allow ballot initiatives which, if passed by a majority of the voters, have the force of law.