Democracy is the people rule themselves through voting. Democracy comes in two flavors. Direct democracy is where the people vote on the laws directly. Indirect or representative democracy is where the the people elect representatives to vote on laws for them.

A republic is a government without a king. The United States, Germany, France, and India are examples of democracies that are republics (also known as democratic republics). Canada, the United Kingdom, Australia, and Spain are examples of democracies that are not republics, as they all have a king. All of these are indirect democracies.

An example of a non-democratic republic would be North Korea.

An example of a country that is neither democratic, nor a republic, would be Saudi Arabia.

BONUS ANSWER!

Some countries have powers devolved to independent subnational governments. This system is called federalism. The opposite is a unitary state. The United States and Canada are both federal democracies, while France and Spain are not. (And to reiterate on the previous point: The United States and France are both republics, while Canada and Spain are not.)

Seriously, we went over this stuff in 5th grade social studies. People only unlearn this stuff when it becomes politically convenient to unlearn it.