"money is coined liberty." --Dostoevsky
Take out a piece of currency and look at it. Chances are that if you live in a western county it will be a printed piece of paper bearing words that indicate it is in fact a bank note. But what does that mean? This term for what is today considered money comes from the origins of printed paper as a representation of value. Governments almost never issued paper money before the eighteenth century; the people who did were bankers. They gave out notes, meaning a piece of paper that promised to pay a certain amount upon the demand of the holder of record at the bank, sometimes at some later date and making a profit by means of fractional reserve.
Now these were not important as a medium for exchange when the current age of banking began. (There have been banks and banking systems for thousands of years, even before coins were invented. Notably in Mesopotamia.) But gradually the bank note became more than a note of deposit saying "the bank shall pay X amount to bearer". People started to sign them over to others, to endorse them just like we do with checks (cheques) today. This was much simpler than going down to the bank to withdraw the money, so instead of large deposits banks started to issue them in set amounts and with elaborate printing to prevent forgery. Now these notes from the bank started to be treated as money by the populace. This was especially true in places where actual amount gold coins in circulation was short (notably the US), because by issuing these notes and then giving gold to the borrower the money multiplier effect came into play. Simply put the borrower did something with the gold, bought something like a business or property and it is likely that at least some of these loans would come back to a bank… That would issue bank notes and loan out the coins again.
Of course there were problems. Bank notes were not valued at their face, whoever accepted them would usually place a premium based upon his confidence in that particular bank and how far off when it could be turned in for actual coins. And banks failed, sometimes quite spectacularly after a run on the bank or because the bank was engaged in fraud, usually involving the issuance of far too many notes. This uncertainty made it hard to do business over long distances.
So in country after country one bank was made the national bank and issued notes that could be used as legal tender and through various means the issuance of private bank notes was restricted or taxed out of existence. This continued until today when national or central banks produced the legal tender for most countries.