Whether it's a presidential or parliamentary system, what makes a modern democracy a true democracy is faithful adherence to a set of democratic principles: the rule of law (constitutionalism), representation based on free and fair elections, and guaranteed rights including freedom of speech, press and religion. By that measure, some countries are democracies in name, but not in practice.
One major difference between presidential and parliamentary democracies is that parliamentary systems have fewer checks on power, since the executive and legislative branches are controlled by the same party. That means that there's generally less gridlock in parliamentary politics, which is great for the party in power, but less so for the minority opposition.
This American style of democracy is called the "presidential model," since the president is the chief executive and is elected separately from members of Congress. As the head of the executive branch, the president also exercises certain powers, like the ability to veto bills passed by Congress, to appoint members to the Supreme Court, and to serve as commander in chief of the military.
That division of power was based, in part, on the parliamentary model that existed in the United Kingdom in the 18th century when the Constitution was drafted. The U.K. wasn't a democracy at the time, but in addition to a monarch (king or queen) it had a powerful Parliament (legislative assembly) composed of representatives at least nominally elected by the aristocracy.