When you begin to delve into the question of what money really is, you must be prepared for some metaphysics. Money, currencies, and other such media of exchange differ markedly in their backgrounds and means of operation, and have changed quite recently into forms that are barely understandable.
For centuries, minted coins not only represented the value and trust of banks, their depositors and eventually nation-states, but also were deemed valuable because they were made from precious metals like gold and silver. These metals are difficult to move around in large quantities, and so banknotes were invented as early as the seventh century in China and brought to Europe in the 13th century. Unlike coins, banknotes were not treated as valuable in themselves since they were simply printed on otherwise worthless paper. Rather, they served as a form of promissory note or IOU that could be presented to the banks that issued them in exchange for their face value in precious metal, coins, or bullion.
In the 20th century, most central banks and governments stopped backing up their currencies with precious metals, and yet banknotes maintain fluctuating values, with some in high demand as media for exchange both domestically and internationally. Dollars and euros are highly regarded and preferred currencies for international commerce, as well as for stocking private bank accounts.