Among the four types of government, monarchy and democracy are reasonably clear-cut, but the ideas of aristocracy (government by the best) and oligarchy (government by the few) have always caused argument. This is largely because "the best" is such a vague and relative concept, and small, elite governments have always tended to think they were the best. This is clearly what the Founding Fathers of the United States thought when framing the Constitution.
The executive branch of government in the United States is clearly a monarchy, a rule by one person, though the president does not have the traditional trappings of kingship. Perhaps the president's most vital role is that of commander-in-chief of the armed forces. This is a role that has traditionally been fulfilled by a single person, at least in wartime, even in republics and democracies.
The judicial branch of government is both an oligarchy and an aristocracy. There are very few justices on the Supreme Court, and they are appointed from a small pool of highly-qualified candidates.
The legislative branch of government is the most difficult to classify. It would be too neat and simplistic to call it a straightforward democracy. Perhaps the best way to describe the United States Congress is to say that the House is a representative (rather than participatory) democracy, while the Senate contains elements of democracy and oligarchy.