A democratic republic is the idea which combines both republicanism and democracy. This means that people can take part in the decisions that affect the way their community is run. There is no monarch. The country could be a presidential, semi-presidential or parliamentary.
Many countries consider themselves democratic republics. Many of these countries are dictatorships where elections are rigged in various ways. For example, in Eastern Europe, countries had Soviet troops stationed after WWII. Elections were organized so that the electorate had no real choice. The consequence of this is that "democratic republic" has become a synonym for "dictatorship.