Direct democracy = popular vote, representative democracy = elected representatives.

China, Russia, USA, France and Switzerland are all republics: two are dictatorships, two are representative democracies, and one is a direct democracy. Canada, UK, Japan and Belgium are all constitutional monarchies, and all of them are considered democratic. Saudi Arabia and Qatar are monarchies, but certainly not democratic.

Democracy is a scale, probably better used as an adjective: the more democratic, the more power is wielded by the people.

A republic is just a non-monarchy. It doesn’t imply anything else about the government of the state in question.