Democracy means rule from the demes, the lowest geographical subunit of the Athenian state.
Republic is from the Roman Res Publica, meaning the public thing. It just means the Roman political order, which has no modern conterpart: it was sort of a truce betrwen older aristocartic and more recent democratic institutions held together by fear of outside threats. It collapsed into tyranny as soon as the old republic triumphed these threats. Therefore everyone from left to right can have a Republic. Its a very popular word that doesn't mean anything except what you want it to be. Its just the government you have to deal with.
Our republic is truely based upon rule from below and we are thus a Democracy in the truest sense. None of the ancients would have any doubt about this if they could see our everyday society and even less so if they could read our constititution. In fact, they would be amazed by the extent of the developement, as no ancient democracy ever extended the rights of citizenship to more than a fraction of the reliable and native male population.
Our constitution is based on an Enlightenment political and philosophical analysis (best word I can think of) of the Athenian and Roman experience, what was good and bad about it, and, since we are an English speaking country, the Englich concepts of Common Law and the English experiences in the their own 1600's revolutionm, whose ideas and values were crushed in England by a restoration of the Monarchy, but largely victorious in the New World colonies. The English King's attempt to establish his Royal perogatives in America was the root cause of the American Revolution.
The European mind could conceive of only four types of government: aristocracy, oligarchy, democracy and tyranny. There may be a few more: odd tribal types (clans, moieties, etc.) and the modern party state, if you do not consider that a type of tyranny, as most do.
Today there are no true oligarchies left (were the rule of wealthy families is codified into law), very few aristocracies, but a lot of wobbling democracies and a lot of true tyrannies.