The treasury is an account holding a country's money. In EU2, there's no money outside of state treasuries, and all money is ducats. All references in the EU2 wiki to "money" should be assumed to be money in treasury, unless clearly noted otherwise.
Seeing the Treasury
For your own country, this is easy. You can always see the amount of ducats in your country's treasury on the status bar at the top of the main window, next to the "coins" icon.
There's no way within the game to find out much money any other country has, except that if you are at war with a country you can go to the peace treaty screen and click on money tribute to determine the enemy's treasury to the next lowest 25d. Each existing country's treasury is written into all saved games, so you could determine it there if you really wanted to know.
Some Uses for Money
Having sufficient money in treasury is a precondition for all sorts of actions. Here is a list of some of the most important ones: