Money Becomes Currency
There are two stories to this. The first is the debasement of coinage and the use of nominal values. The second is the modern form currency being claims on money.
Coins
Originally, coins were valued for their pure metal content, but as civilisations developed, certain standardised coins emerged that were relied upon by the people to contain fairly exact amounts of pure metal, much like well established coin mints today.
Most modern currencies were demoninated and defined by weight. Dollars, originating from Joachimsthalers (Bohemia, 1518) were defined as 371.25 grains (24.057 grams) of pure silver. A pound sterling was defined as a pound of sterling (.925) silver. The word "peso" means weight. So, when you spent a dollar you were really spending a weight of silver. Unfortunately it wasn't long before these weight distinctions were largely forgotten and the names took on a significance of their own.
At these various points in history, the leaders of the day often seized opportunities to defraud the people, none more so than the Roman Empire on a number of occasions. By placing a nominal value on the coin and then shaving them, drilling holes, cutting pieces off and smelting with more common metals, the Romans could create more currency than the available gold and/or silver would have otherwise allowed. This increase in the supply of money transferred wealth away from the people to the empire in a way that most people couldn't detect - at first. Eventually the people recognised what was going on and would only accept these debased coins at a discount. In some cases they would refuse them entirely and prefer instead foreign coins whose metal content they could still trust. Rather than realising that the game was over, the governments often chose a more totalitarian path.
For example... Emperor Diocletian in the third century AD, in an effort to fund domestic spending and empire building continued this currency debasement. When the people began to lose faith in these debased coins, he actually issued price controls and attempted to force the people to accept their "legal tender" coins by threatening them with the death penalty if they didn't comply. As a consequence, rather than lose money by accepting debased coins or risk death by selling goods at their true value, merchants simply stopped trading, causing an economic depression. Without realising it, Diocletian had discovered one of the problems that communism in the 20th century would encounter; the lack of incentives destroys economic production.
Another effect of the separation of nominal and real values resulted in something called "Gresham's law - bad money drives good money out of circulation". People inevitably spent the overvalued coins instead of the real articles. In fact, the real coins were often taken out of circulation, melted and sold as bullion for a nice profit. Nowadays most countries have laws against destroying coins. In the UK, it is illegal to "deface" anything with the Queen's image on it - so the people are prevented from destroying coins. The money in the UK has been so divorced from any real value it hardly matters, except in the case of pre-1991 copper coins that still circulate. A 2p coin at current copper prices is worth about 4p! When the UK government took the last silver out of any coins, the 50% silver coins (shillings, florins etc.) that weren't caught in the government net, were quickly hoarded by the people. These now trade on ebay for perhaps 25 times the face value. The government itself makes a nice profit out of this process - they collect the 50% silver coins, sell the silver and replace the coins with stainless steel tokens. This process is called seigniorage.
A quick look at the history of money in most developed nations shows a similar story - pre-1965 US Silver Coins (except Rooselvelt Dimes), US Kennedy Half Dollars (1965-1970) and US Jefferson Nickels (1942 (partial)-1945) in the US for example.
Claims on Money
I'm going to tell you an allegorical story now about medieval gold smiths. Rather than being a story of historical accuracy it serves to explain how modern banks were formed and how fraud became entwined in the system. The story goes something like this...
When gold and silver coins circulated as money people were at risk of having their money stolen from their houses and/or person. The realised that the goldsmiths of the day who fabricated jewellery dealt with a lot of gold and silver and had much better security arrangements in place; secure vaults and armed guards. So, the people asked the goldsmiths to keep their money safe for them, which the goldsmiths agreed to for a fee. They effectively became money warehouses, holding the gold as a bailment for safekeeping. In return for this safekeeping they got to charge for these services. The first deposit banks had evolved. People handed over their gold and silver coins for safekeeping and the goldsmith handed them back a receipt of deposit, in much the same way a dry-cleaner gives you a claim ticket for your clothes.
These paper tickets began to circulate as money. They were the technological advancement of the day. They were convenient and since gold and silver coins were fungible, it meant that trade could be expanded over long distances where goldsmiths had agreements to accept each other's receipts.
These "banks" soon entered the lending business. They were perfectly positioned to lend the peoples' money to various entrepreneurs. This was a incredibly valuable service for both depositors and lenders. If a depositor was willing to encumber some of their funds, they could be paid for lending their money to someone else without being the only lender taking all the risk. Loan banking allowed many participants to take a small risk in many different and varied ventures. The banker in the middle received the spread between what they charged to borrowers and what they paid to lenders. Loan banking effectively built the modern world, funneling capital obtained by saving to businesses who innovated - creating wealth and employment in abundance.
However, the bankers realised something incredibly important and, as it turned out, very profitable. Only very rarely did anyone actually redeem the gold or silver. The bills either circulated as money or, in the case of larger transactions, the ownership literally changed hands by a simple ledger entry. Importantly, in both cases the gold and silver remained in the vault. In a similar ploy to debasing coins, the bankers spotted a chance for profit. They began to issue more receipts than they had backed by gold and silver in their vaults. They could make loans with newly-created, unbacked receipts and charge interest on them. As the loans matured and were paid back, the risk that someone would discover they were unbacked disappeared and the bankers were left with a tidy profit. Had they done this in very modest amounts, the public would have been none the wiser. However, as they flooded the market with these unbacked receipts, substantially raising the supply of money in circulation, it flowed into all sorts of goods, bidding up their prices dramatically. As people started to lose confidence in the currency they tried to redeem their receipts for the gold and silver they supposedly represented only to find out to their horror that there were more claims than could be paid out with the available bullion and/or specie. The first bank run had occurred.