Deposit Banking
Warehouse Receipts
Origins of deposit banking as a warehousing service, where individuals deposited their gold in a warehouse that provided a receipt that could be presented to collect the goods. The warehouse makes a profit by charging a fee for safeguarding the depositors' property.
Deposit Banking and Embezzlement
The danger is embezzlement: the warehouseman could steal their goods, or some portion of them (especially if they are commodities), and that eventually, receipts would exceed the amount of gold in storage. Also, the warehouseman could print counterfeit receipts and put them into circulation.
Other than theft of property from the owners, the problem with embezzling gold or counterfeiting certificates is that the amount of "money" in circulation (assuming the gold/notes are put into circulation) is increased by the amount of receipts (in excess of actual gold), decreasing the value of money, causing prices to increase in the entire market.
A historical case (Foley, 1848) arose when a bank loaned out funds that belonged to depositors, yet the receipts remained in circulation as money. This case upheld their right to do so, and ruled that the receipts were merely debt markers indicating the amount of a bank's assets an individual owns. Yet, they still continued to be used as currency.
As a result, the money supply could effectively be doubled: the bank could provide a receipt to an account holder for a pound of gold that would be circulated as currency in the market - and, at the same time, loan out those same pound to others, to be used in the market as currency. Hence, based on one pounds of gold, two pounds are in circulation.
It gets more complex when the borrowed funds are deposited yet again. An individual deposits a pound of gold and receives a receipt, another person borrows it. This second person deposits the gold in another bank, receiving a receipt, and that bank loads the pound to a third party. Three pounds are now in circulation.
Foley remains the basis of American banking. Only in the case of safe deposit boxes is the bank required to hold the actual goods for which it issues a receipt. Otherwise, the bank may loan out deposited funds.
Fractional Reserve Banking
The danger in the Foley system is that, if the entire amount is lent out, the bank will have no gold to redeem receipts, hence its receipts will be regarded as worthless in the market. Economically, this means that the supply of currency would revert to reflect the actual amount of gold (which is good), but this outcome is undesirable for the bank.
And so, the bank must retain a reserve of gold so that it is able to honor its receipts and thus maintain their value in the market. The bank must predict the portion (fraction) of gold on deposit that it expects customers to redeem, and can "safely" loan out the remainder.
This is fractional-reserve banking, in which the amount of gold on hand is lower than the number of receipts outstanding. Economically (and objectively), this is a form of fraud, and its effect on the market are identical to counterfeiting. Legally, based on the Foley precedent, it is entirely permissible to perpetrate such a fraud.
Bank Notes and Deposits
A bank note is a receipt for funds deposited in the bank. These notes, created in various denominations, are transferable and are exchanged as money in the market. In effect, they are paper money.
While the bank notes were based on gold, all was good and well - but when governments replaced gold with paper money, this became an issue: there would be no incentive for customers to deposit government-issued paper in exchange for bank-issued paper.
The solution for banks was the deposit account - in which the amount of money deposited is recorded in a ledger, and the owner is given the privilege of writing checks that others may redeem (fundamentally, the same as receipts were redeemed), with the added convenience that checks can be written in variable amounts, whereas the bank notes were in specific amounts (and you had to make change).
Fundamentally, the practice is the same - both bank notes and checks function as receipts, in that they are accepted as currency by the market.