It helps to bring at least one additional term into the discussion: that of a monarchy.
As Wikipedia states:
A monarchy is a form of government in which a single person holds supreme authority in ruling a country, also performing ceremonial duties and embodying the country’s national identity.
Monarchies are traditionally associated with royals, i.e. kings or queens of some sort. The counterpart to a monarchy is typically seen as a republic in which the head of state does not have the same role in embodying the country’s identity (compare the status the Queen of England has with that of the president of Germany although both hold pretty much the same actual power nowadays).
While it may be difficult to distinguish a powerful, unelected, dictatorial president from an absolute monarch one would typically use the titles they give themselves (this means that North Korea is typically considered a republic rather than a hereditary monarchy).
Using the not-so entirely clear definitions, the United States are a republic because their head of state is a president (not a king or queen) who is never himself (or herself) seen as embodying the nation or country itself.
The question whether something is a democracy is on a different scale: it concerns who has the power to determine the government. In the original Greek, δῆμος (demos) means the people while κρατός (kratos) means the power (to rule). Thus, a democracy is characterised by the people (or a significant part of it) effectively and powerfully choosing how the state is to be run.
In most democracies this is achieved by an elected parliament where it is assumed that a member who does not vote in the way the electorate wants will not be reelected. Obviously, a number of caveats apply so the mere existance of a vote does not suffice to call a system democratic; but there are inherently democratic systems that do not rely on voting, e.g. if a small village (or an entire Swiss canton) decides all or most of its issues in a congregation of all people living there.
As the people of the United States elect their president, the House of Representatives and the Senate as well as a number of other positions, and since there is typically an effective choice at some point in each process (even safe seats can change against the will of the incumbent thanks to primaries, although these are muddier waters) the United States are a democracy.
It is important to note that the terms monarchy and democracy are not mutually exclusive. The United Kingdom, for example, can be classified as a democracy as the elections to the House of Commons are typically the decisive factor in determining national politics in the next five years while the Queen as head of state remains a monarch.
But it is not necessary for the position of a monarch to be hereditary; it is entirely possible to devise a system in which the people elect their monarch for a certain period while the holder of that position is considered a monarch in the full sense of the word. Obviously, indirect elections of monarchs are also possible and may be happening in our current world, depending on your precise definition.