Most modern economies use a counter-intuitive model of banking called "fractional reserve banking." It is counter-intuitive (and some people would say wrong) because it allows banks to lend out money that it tells depositors is available at any time and essentially involves private banks in money creation. It also creates the possibility of mass instability through bank runs that tend to be mitigated through government regulation and insurance (some would say government subsidy of banks).
This tutorial explains how fractional reserve lending works and outlines the good and bad. It also talks about the alternative of full reserve banking.
This short tutorial explains how we measure how much "money" there is out there. As we'll see, this isn't as straightforward as counting dollars in people's pockets, especially because there are multiple type of money.