I've read various pieces arguing that the USA is not a democracy, but a (fill in the blank, ex., a Republic). Perhaps there is no perfect fit, but which form of government does the USA most closely match?
The definition of democracy varies, but for most definitions used, yes, the USA is a democracy.
Wikipedia's article on Liberal Democracy defines a liberal democracy as follows:
It is characterised by fair, free, and competitive elections between multiple distinct political parties, a separation of powers into different branches of government, the rule of law in everyday life as part of an open society, and the equal protection of human rights, civil rights, civil liberties, and political freedoms for all persons.
Though Gerrymandering and geographical concentrations of like-minded voters make elections less and less competitive, the United States meets these criteria. Elections are not, by and large, systematically rigged in favor of a single party, open to almost every adult American citizen (with the notable exception of some felons in some states), and there are multiple political parties that endorse different policies. Power is separated into an Executive, Legislative, and Judicial branch, and the Constitution guarantees certain civil rights and liberties.
The article also lists a number of Democracies, noting that they can be compatible with other forms of government:
A liberal democracy may take various constitutional forms: it may be a constitutional republic, such as France, Germany, India, Ireland, Italy, or the United States
However, democracies need not be liberal democracies. We can define a democracy more narrowly, as
a form of government in which all eligible citizens are meant to participate equally – either directly or, through elected representatives, indirectly – in the proposal, development and establishment of the laws by which their society is run.
At these criteria are the subset of a criteria for a liberal democracy, than any liberal democracy is also a democracy. If we can consider the United States a liberal democracy, we can consider it a democracy.
One can argue that the United States does not meet all these criteria perfectly. Many felons are not allowed to vote, and per-state representation in the Senate means that not every voter has equal say in how laws are made or conducted. However, nobody's perfect. Keeping this in mind, some organizations have chosen to rank countries by how democratic they are, taking into account multiple criteria. These organizations' definitions of democracy vary, but they tend to consider the USA a democracy.
The Economist Intelligence Unit gives America a rating of 8.11 out of 10. It considers any country with a ranking better than an 8 a "full democracy". By This definition, America is a Democracy. Freedom House has a list of electoral democracies which includes America, and also lists America as a "free" country.
It should be noted that a government can be a democracy and a republic, or even a democracy and a monarchy. The principles of democracy imply very little in the formal structure of a government. You can have a democracy that involves direct voting on certain issues, like in California, or you can have a monarch with Constitutionally limited powers, like in the United Kingdom, or a Republican Democracy without an elected head of government and no monarch, like the United States.
The United States is a Representational Democracy. The people have no direct say in government, but have the ability to freely vote for those who do have direct say (the people's representatives).
This is contrast to a Direct Democracy (which is also sometimes called just a "Democracy" or a "pure democracy"), where every citizen is entitled to a vote on every issue (such as ancient Athens), and a Delegate Democracy where citizens may temporarily give their votes to a chosen delegate who can vote them as a bloc (similar to a corporate board). There are other types of democracy it's also not.
It is also a Republic, which is a term that can be applied to any country which is democratic but not a direct democracy. Technically, this makes the US a Democratic Republic, but the countries which currently use that term do not use it in that way, and that usage isn't common.