If we live in a democracy with presidents, what is the government with the kings and queens?
A monarchy is a government with kings and queens. Monarchy is the oldest form of government in the United Kingdom. In a monarchy, a king or queen is Head of State.
England= Constitutional Monarchy
The US is actually an oligarchy
If the Sovereign reigned with power I imagine it would be a Monarchy. Most modern Monarchs have limited power and are controlled by democratically elected parliaments. So they are democratic monarchies. In Australia, our constitution names the English monarch as our head of state, and she (and her representatives) have limited powers. We are a "self-governing democracy". Some of the Middle-Eastern states come into a category all of their own. They are kleptocracies: governments of thieves.