A republic is the opposite of a monarchy. In a republic, leaders do not inherit their position, and they are not chosen by their predecessors or their predecessors' families. A republic may be a democracy, either direct or representative, or it may be an oligarchy or a dictatorship, with a ruling elite choosing its leaders in democratic or quasi-democratic fashion.
Not all republics are democracies though. North Korea, Burma, Iran, (United States of America really? er well), etc term themselves republics but cannot be considered democratic. Communists have made a big show of being democratic, but their elections have had only one party's candidates, and we all know which party that has been.