Republics and democracies both provide a political system in which citizens are represented by elected officials who are sworn to protect their interests.
In a pure democracy, laws are made directly by the voting majority leaving the rights of the minority largely unprotected.
In a republic, laws are made by representatives chosen by the people and must comply with a constitution that specifically protects the rights of the minority from the will of the majority.
Want to learn more? This video does a great job of explaining the difference.