The Supply of Money
The supply of money is the total number of currency units in the economy.
What Should the Supply of Money Be?
Unlike other goods, the supply of which is often dependent on factors beyond a central control, the supply of currency is controlled by central government, which may seek to influence the supply of money to effect social good.
In the long term, an increase or decrease in the money supply confers no social benefit (if you doubt the amount of money, prices will double, and the net effect is zero). There may be a certain short-term benefit to those who act upon the change before others catch on and bring the market to equilibrium.
Fundamentally, Rothbard argues that the supply of money should be left to the marketplace, just like the supply of other goods. He concludes that the "problem" of a proper supply of money is not really a problem, nor will manipulating the amount of currency be likely to have a beneficial impact in the long term.
The Supply of Gold and the Counterfeiting Process
When currency is based on a gold standard, the only thing that can legitimately affect the supply of money is the supply of gold. He seems to be arguing that the only point in gold mining is to replace the money that is consumed in practice (in addition to backing currency, gold is used).
Counterfeiting is a method of creating currency without increasing the amount of the commodity (gold) on which the currency is based. The effect of counterfeiting on an individual scale is short-lived: once the fraudulent currency is discovered, someone has lost value, but the market as a whole is unaffected.
Counterfeiting on a large scale (governments creating currency that is not backed by gold) is the equivalent of debasement - it causes the unitary value of currency to decrease, hence increasing prices.
Government Paper Money
Paper money as a facilitator of government counterfeiting. Originally, it could be exchanged for gold. Governments could not easily counterfeit money because there was the threat that, if there were a panic, those who held it would redeem it for gold, and the fraud would be discovered.
Once taken of the gold standard, i.e., getting the public to accept the paper currency backed by fiat as if it were real money (backed by commodity), government can more freely counterfeit, and alter the supply of money, hence prices across the entirety of a market.
Hence gold coin was taken off the market by Roosevelt in 1933, silver in 1965 by Kennedy, to fully divorce the American dollar from its relationship to any commodity.
The Origins of Government Paper Money
Historical perspective on the manufacture of paper money - the difficulty of creating reliable cash (genuine, difficult to counterfeit, etc.). Seems largely trivial to the topic of banking, except to note that banks once created "paper money" by making deposit certificates transferable, but the practice of banks issuing paper money redeemable against their stock of gold has been outlawed.