How is a republic different from a democracy?

Republic, roughly, refers only to the rule of law, or that all members of society (specifically, citizens) are accountable to its rules. Autocratic polities can be republics (i.e.) a polity might not determine who rules by the people, but it might still hold everyone to the rule of law. A good example of this might be Singapore).

Democracy refers only to the rule of the people. It is possible to have democracies that are not republics (e.g.) ancient Athens).