Guys. This is my field; I'll be happy to clear this up.
The United States is both a democracy and a republic, because a republic is a form of democracy.
A republic is a form of government in which power is split among many representatives of the people, because the government is public—literally, res publica means "public thing." We are a republic because legislative power is split among 535 elected members of Congress, because every branch has checks and balances against each other or even within themselves (such as with the 25th Amendment), and because all of our officials are either elected or appointed by those who were. Other nations, by contrast, may be absolute monarchies, where the government is a private thing held by the monarch, and they have final word over everything.
A democracy is a form of government in which the people (demo-) have the power to make change. We are a democracy because we have elections, which give us the power to put the people we want in office. Other nations, by contrast, may be autocracies, where one person (auto-) has the power to make change, or theocracies where the church (theo-) has the power to make change. Because a republic relies on elected officials, it is by definition a type of democracy.
There is another type of democracy, though, called the "absolute" or "direct" or "Athenian" democracy. In that non-representative system, every citizen votes on every proposal, every time. A lot of people confuse the term "democracy" to mean this subtype, and as you can imagine, it would be entire unfeasible for a massive nation of 330 million people. We are definitely not an absolute democracy, but we still are a democracy nonetheless.
I hope that helps.