What is public coin?
A public coin is any exchange commodity for which the total supply is public. It is any money in which everybody knows how much there is. Compare this to a "publicly traded company". Such companies have a list of shares that is public, which makes the supply of shares public knowledge. A public coin is analagous, in that it has a money supply which is public. It represents a public commons, of which anyone is free to participate.
Satoshi's bitcoin is a public coin. However contrary to common belief, bitcoin was not the first public coin. The Rai, as used on the Yap islands, predates it. However the Rai coin was not globally used, nor electronic, nor so easy to use as bitcoin.
The advantages of public coin were laid out by Wei Dai in his proposition called B-money (1997). This system was later implemented by Satoshi (2008) using proof of work. Quite simply, monetary supply fraud such as counterfeiting, hidden issuance, and private dilution, is completely eliminated from such a system.
Today there are thousands of public coins in use, of which bitcoin is the most common, and they are the preferred way of doing business amongst those willing and able to wield political power in the form of monetary choice. After the dismal record of the counterfeiters and fiat issuers and what they did with their wealth during the 20th century, this is a most promising development.