Mmmm, it generally works like this.
In each country, it is generally illegal to print your own money. Normally it's called counterfeiting.
Usually, only a specific government body can print money. This body is called the "central bank". Despite the name, it is not a banker like, say, Citicorp or Deutsche Bank or JP Morgan. You can't go there and get a mortgage or apply for a credit card, for instance.
The central bank "feeds" money into the system, through the banks. The central bank also "sucks" money out of the system, when there's too much money leading to inflation.
Yes, money is printed and stored in vaults. A retail bank that wants plenty of currency notes could simply, say, sell government bonds worth $500,000,000 back to the central bank, and the central bank could pay by using the notes that it has printed.
The notes are then delivered by truck, with security guards etc etc, to the retail bank. Which can then take the money and load it into its ATMs around the city.