Contrary to right-wing propaganda, a Republic is indeed a form of representative Democracy in which ordinary people elect officials to represent them in a legislature or parliament. Simply put, a Republic is the "what" and a Democracy is the "how." It's been that way since
the Roman Republic was established in 509 B.C.
The notion that Democracy is a mob-rule involving direct citizen voting on every issue is a distortion spread by idiots like Tom
Tancredo and radical ideologues like Pat Buchanan. They don't like the term "Democracy" because it reminds them of the Democratic party.
The opposite of a Republic is a Monarchy,
not a Democracy.