Fractional Reserve Banking and Bitcoin
Fractional Reserve Banking with Bitcoin is possible. There is no fundamental difference between classical currencies and Bitcoin as it applies to banking. Banks will still be free to take in bitcoins and present them to customers as "available for withdrawal" while still lending most of those bitcoins to a different customer for a profit. Some of those bitcoins will be held in reserves in case of a bank run. It will be up to the bank to hold a sufficient supply of reserves in order to prevent insolvency in the event of a bank run. Central banks were established to enforce reserve requirements and so, with Bitcoin lacking a central bank, some banks will almost surely collapse, taking their customers' deposits with them.
Conventional banks in the United States guarantee that account holders can withdraw 100% of their dollars based on their "word" and the fact that they are backed up by the FDIC. This program insures depositors up to a certain amount (currently $250K USD per depositor). The FDIC is widely known to have reserves sufficient to cover only a very small fraction of the total deposits it insures though the FDIC itself can be considered to be backed up by the US Congress in the event of its insolvency. After politically desired, the FDIC's role could be extended to insure Bitcoin banks and establish a minimum reserve requirement. Such a change would only happen after public outrage occurs after the inevitable collapse of major Bitcoin banks.
The Monetary Base of Bitcoin is limited to 21 million. But because Fractional Reserve Banking is possible, the money supply of bitcoins (which includes demand deposits) can greatly exceed 21 million.
Fractional reserve banking arises when issuers of money substitutes create more substitutes than the reserves they have. The process is described in more detail in Wikipedia article on Fractional reserve banking:
As most bank deposits are treated as money in their own right, fractional reserve banking increases the money supply, and banks are said to create money.
Practicing fractional reserve banking without issuing money substitutes is not logically possible.
With fiat money, the base money are the reserves commercial banks have with the central banks, and the substitutes are the account balances of the customers of commercial banks. With gold, the base money is the gold bullion/coins stored at the bank, and the substitutes are bank notes, cheques, account balances or other instruments the commercial banks provide to the depositors.
The reason why substitutes are accepted as if they were the base money is that the base money has in some circumstances high transaction costs (for example, gold might be too heavy to carry around, or the buyer and seller are not at the same location and want to perform the exchange electronically), or are not legally permitted (normal people are not allowed to obtain central bank reserves). This creates a demand for forms of money which have lower transaction costs. With gold/fiat, this requires the creation of money substitutes.
The situation with Bitcoin is different, because other forms can be created without substitutes, for example Casascius physical bitcoins or BitBills. Bitcoin in its "classical" form is the equivalent of a bank deposit. An issuer of a Bitcoin substitute, before they even think about practicing FRB, would need to find a way of creating demand for these substitutes. This is problematic, because such substitutes are incompatible with the Bitcoin network, and need to compete against not only Bitcoin, but against other currencies, payment methods and services. Currently, the only examples of such substitutes are account balances on exchanges, and most e-wallets (Strongcoin being an exception, using client-side encryption, the balances of their deposits are just another way of storing a Bitcoin wallet, so they are not substitutes). Even in these examples, Bitcoin substitutes do not circulate outside of centralised systems, so performing FRB with them has a limited impact on the money supply and is very risky.
If in the future, P2P exchanges and distributed wallets are available (both have been suggested already at bitcointalk.org forums), this would decrease the demand for Bitcoin substitutes even further.
Historically, in all the situations where fractional reserve Bitcoin substitutes were produced, this resulted either in a voluntary elimination of the excess substitutes (Mt. Gox hack from June 2011), bankruptcy (the demise of mybitcoin) or a new investor bailout (the demise of bitomat.pl and subsequent takeover by Mt. Gox). Here we have empirical evidence that FRB with Bitcoin is possible.
In face of the lack of demand for Bitcoin-substitutes, it is doubful whether FRB with Bitcoin can be conducted in a profitable manner and have a similar impact and scale than with fiat/gold.