Is monarchy possible in a democratic country?
Monarchy is possible in a democratic state, but of course this depends on the monarch and the democratic body. The best example of a monarchy-democracy blend would be Great Britain. Elizabeth II reigns as a figurehead. She has little actual political power. The real power of state resides with the Prime Minister and the Houses of Parliament, of which the Lower House is elected by the people.
Monarchy and democracy used to be on opposite ends of the spectrum, however--there was an idea called "divine right" where the king/queen ruled because he/she was God's secular sovereign on Earth. Under this system there might be an elected body, such as the Duma under Nicholas II in czarist Russia, but it was only an advisory committee and the ruler could do whatever he/she wanted. This theory has largely fallen out of favor, however, due to many countries enacting reforms to give the people more of a say in their political futures. Also, absolutist monarchies have often ended badly in history when the royal hereditary line fails to produce an heir or the heir is incompetent to rule.