How is it possible for a country to be democratic but not a republic?

Answer 1

The word "democracy" comes from the Greek meaning "rule by the people" or power to the people and the word "Republic" comes from the Latin meaning "of public affairs;" so they are so close in original source as to be hard to separate except by degrees.

Republics can vary in considerable ways in regards to how the country is run. Basically, as long as the country is not a Monarchy it is a Republic. Monarchies can have democratic elements and be more and more democratic to the point where the Monarch is a figurehead or performs the role of President in a Parliamentary form of government. One way this situation is described is as a "Constitutional Monarchy" wherein a written constitution describes limits on the monarchy and expresses powers to the citizenry.