I am recalling a slogan of the John Birch Society from back, I think, in the early 60s:
This is a republic, not a democracy. Let's keep it that way!
Without starting partisan bickering, what are the arguments for considering the USA to be a republic vs a democracy?
Based on clarifications in the comments, I think it's safe to start on Wikipedia:
Its page on republic starts with:
A republic is a form of government in which the country is considered a “public matter”, not the private concern or property of the rulers. The primary positions of power within a republic are not inherited, but are attained through democracy, oligarchy or autocracy. It is a form of government under which the head of state is not a hereditary monarch.
Its page on democracy starts with:
Democracy is a system of government where the citizens exercise power by voting. In a direct democracy, the citizens as a whole form a governing body and vote directly on each issue. In a representative democracy the citizens elect representatives from among themselves. These representatives meet to form a governing body, such as a legislature. In a constitutional democracy the powers of the majority are exercised within the framework of a representative democracy, but the constitution limits the majority and protects the minority, usually through the enjoyment by all of certain individual rights, e.g. freedom of speech, or freedom of association.
The definition of republic certainly applies to the United States. The country isn't owned by a select group of people and the primary positions of power are determined through elections.
The definition of democracy also applies because citizens vote to exercise power. It is not a direct democracy because people vote for candidates, not specific issues. In particular, that means it is a representative democracy. It is also a constitutional democracy because individual freedoms are protected by the constitution.
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.
It's important to understand the U.S. has a lot of weird terms that don't mean what the rest of the world thinks they mean. The quoted phrase is a bit more archeic use of "Republic" in the United States, where it's best understood as a Representative Democracy. This stems from the fact that back in the Day, the United States was one of the First Representative Democracies that was also a Republic... and it did it in the time where this system of government didn't really have a name (Constitutional government was not a well known thing in the late 18th century and the United States was the first nation to adopt a constitutional republican form of government and the U.S. Constitution served as a model for most other governments that would adopt a representative democratic form of government.). With all this in mind, it's easy to understand that the term "Representative Democracy" was not a thing when the U.S. created it and they referred to the government as a Republic, not a Democracy, because they understood Democracy to be Direct Democracy... which the founding fathers loathed as much as they loathed a monarchy.
The question here is best rephrased as "The U.S. is a Representative Democracy, not a Direct Democracy." to understand the idea being conveyed. So to answer the question, yes, the U.S. is a Republic (in the sense that we do not have a Hereditary Monarchy) and Yes we are a Democracy (in the sense that we are a Representative Democracy). As it was used in context, Republic is using a unique American definition that basically means Representative Democracy (Which we are) vs. a direct Democracy (which we most certainly are not).
