"Islamic banking" refers to banking done according to the principles of Sharia, the holy law set forth in the Qur'an. Specifically, the Qur'an forbids riba, or the charging of interest in exchange for a loan, which was also the practice of the Christian church during medieval times. The problem is that modern banking systems are built around interest: the time value of money is a central concept of finance which states that a certain amount of money is worth more now than at some point in the future, because if you have that money now, you can use it in the interim to invest and earn more money. Thus, essentially, if you have extra money that you could loan out, it makes little sense to do so unless you're receiving interest for it. What sets Islamic banking apart, then, is the strict adherence to religious restrictions on usury and what is done instead to build a modern, functioning financial system.
Usury
Usury traditionally referred to making money by handling money — specifically, to charging fees for loans in the form of interest or to charging to change money. The scriptures of many religions forbid usury — Jews are forbidden in Exodus to lend money at interest to other Jews1, and Deuteronomy elaborates by explaining it as an injunction against earning money from a fellow's hardships.
While Christians don't follow all of the laws of the Hebrew Bible, the position of the Catholic Church in the middle ages also condemned usury; in Dante's Divine Comedy, userers were condemned to the seventh circle of hell along with blasphemers and sodomites. Shakespeare's The Merchant of Venice depicts the moneylender Shylock as something of a Jewish caricature, greedy and resentful and demanding the unfair repayment of a pound of flesh in exchange for a defaulted loan. But at the end of the play, he gives up his ill-gotten wealth and converts to Christianity.
It's clear, though, that restrictions on usury in Judaism and Christianity pertain to situations in which a person earns money by taking advantage of another's misfortune. Religious texts are rife with examples in which usurers are held in the utmost contempt. But as modern finance systems began to develop, many of these restrictions loosened. During the middle ages, European Jews frequently went into business as bankers serving the Christians around them (as with the above example of Shylock), since this didn't violate the commands of the Torah. But eventually, in Europe, Biblical injunctions on usury were understood as condemning the practice of taking advantage of those in need; as a result, the term "usury" was applied to loans with abusively high interest rates and banking systems built around interest were accepted, given their importance to the nascent capitalist economies of post-Renaissance Europe. John Calvin, an important early figure in Protestant Christianity, actually wrote a defense of charging interest; the Roman Catholic and Eastern Orthodox churches began to recognize usury only when high interest rates were charged.
The Qur'an contains references to usury just like the Christian and Jewish scriptures. One of the most succinct references is in Al-'Imran 3:1302, which makes specific reference to compound interest; Qur'anic statements condemning usury are explicit, but they are not offered with the same rationale as is found in the Jewish scriptures, which specifically contrasts usury with freely taking in needy relatives. Perhaps the fact that the Qur'an doesn't offer Allah's reasons for the ban on usury explains why Islam never loosened its strictures on moneylending to apply only to predatory lending in particular. Whatever the reason, while parallel restrictions on lending at interest have been essentially forgotten in Judaism and Christianity, they are alive and well today in the Islamic world.
1Exodus 22:25 "If thou lend money to any of my people that is poor by thee, thou shalt not be to him as an usurer, neither shalt thou lay upon him usury" (King James Version). Note that "my people" is understood as a specific reference to the Hebrews, a matter made more specific in Deuteronomy.
2Al-'Imran 3:130 "O you who believe, you shall not take usury, compounded over and over. Observe God, that you may succeed."
The Case for Usury
I've made reference to interest as key to modern financial systems and thus an important part of how the economy works. Let's address this further. Economically, interest is the return on capital. Capital refers to money (and physical goods that can be bought with money) invested in a business in order to achieve later gains. Interest thus can refer to the return on a normal investment; traditionally, this isn't usually considered usurious because there is a risk associated with investment.
Lending money for free is a bad bet. It's one thing to offer a no-interest loan to a person in need; this seems to be the situation addressed in the Old Testament. But having money now is worth more than having money later; under a classical economic analysis, it makes little sense to lend money for free when the money could instead be used as capital and invested for a return. Not only is there a loss in terms of opportunity cost, though; inflation guarantees that an amount of money in the future is quite literally worth less than the same amount today. Further, lending money is associated with liquidity risk, the risk that you might not have enough money for yourself should the need arise. Other risks exist, such as the simple risk that the person being lent the money might not repay. Since some risk is involved in lending money, and since lending out money means incurring an opportunity cost and losses from inflation when it's not invested, charging interest on loans makes sense on a strictly economic basis.
Indeed, without the possibility of charging interest, it's hard to imagine how a lot of basic economic transactions could take place. How could someone who wasn't wealthy enough to have hundreds of thousands of dollars on hand buy a house without a mortgage from a bank? Because of the risks associated with lending money, any bank that decided to offer loans without charging interest would quickly go out of business. The ubiquity of loans in our daily lives would be a significant problem were we to give up charging interest; further, there would be little reason to leave money in savings accounts — essentially, of course, a loan to your bank — which would mean that banks couldn't then invest that money and a significant portion of the capital that drives economic growth would disappear.
Islamic banking
So, since banks in the West operate on the basis of charging and paying interest, how do the modern (and extremely capitalist) economies in the more developed parts of the Islamic world do business? The answer is complicated. Many different mechanisms are used to replace familiar banking concepts, but essentially, in most cases, the money that might have been tallied as interest were the same transaction done in a Western bank is instead written into the contract in some other way. In fact, some Muslims consider Islamic banks to be engaging in legal trickery to hide the fact that they charge and pay interest. Most economic analyses would probably agree — compensation is being made for the time value of money, and that is the definition of interest. Nevertheless, banks exist that operate under the principles of Sharia and do not consider themselves to be collecting riba.
An example of how Islamic banks provide services without charging interest is murabaha. The term refers to a practice in which a bank buys an item or piece of real estate and resells it to its ultimate purchaser with a mutually agreed-upon profit for the bank, allowing the purchaser to pay in installments without technically paying interest. Since there is of course no Qur'anic injunction against reselling an item at a profit, murabaha allows the bank to be compensated for the time value of the money spent and the risks associated with lending without technically violating the religious ban on riba. Minor differences between Islamic banks and conventional banks crop up because the bank owns the property in the interim and assumes the loss should something happen to it.
A similar service is ijara wa iqtina, in which a bank buys an item and leases it to the purchaser, who agrees to buy the item at the end of a pre-agreed upon period. Rental payments include an amount towards the value of the item, so that the final payment is generally a token amount. Thus a profit is earned by the lender but technically interest is not collected. Interest is not payed on savings accounts in Muslim banks either, though some banks may give a gift in appreciation for the loan from the customer, usually in the form of periodic deposits based upon the bank's profits; this gift is not guaranteed and thus is not considered to be an interest payment.
Another unique practice of Islamic finance is the concept of sharing risk. Instead of providing normal loans to businesses, practices very similar to venture capital partnerships are frequent. A partnership is formed between a bank and an entrepreneur, with an agreement providing for the sharing of profits or potential losses. Such loans are provided not only to start businesses but also for reinvestment in a business; in the latter case (like the former), the profit to the bank is tied to the business's profits. Thus risk is borne by all participants in such a partnership. This could be held as the basic principle underlying Islamic banking &mdash: in not-Islamic banks, fixed -interest loans can be given without risk to the lender, even if the venture fails. Loans that are acceptable under Sharia, on the other hand, involve a degree of sharing of profit and risk.
Islamic investment services are common as well. The Qur'an forbids earning money by investing in unethical businesses, so Islamic banks and investment firms invest only in ethically acceptable ventures — those that don't involve, for instance, alcohol, pornography, gambling, or other things forbidden to Muslims. Ethical investment is increasingly common outside of Islam nowadays, but there is no religiously acceptable alternative for Muslims.
Sharia advisors
Islamic banks hire Sharia scholars for the purpose of supervising the bank's activities to ensure that they are in compliance with Islamic law. Committees of scholars or independent consultants are hired to establish rules for specific transactions and financial products and then monitor the bank's practices to make sure that they meet the agreed-upon rules.
Status
The first modern Islamic bank was the Dubai Islamic Bank, established in 1975. Previous experiments with smaller Islamic banking institutions occurred in Egypt between 1963 and 1967, but these were simple savings institutions and they operated covertly due to the government's overt hostility towards anything it perceived as Islamic fundamentalism. The Dubai Islamic Bank and over two hundred other such institutions are fully modern banks, and various such banks exist in quite a number of countries, including the United Kingdom and the United States.
Conclusion
Muslim scholars have argued that the modern attitude of Christians and Jews towards usury — defining it as only the charging of abusive or excessive interest — is a rationalization. Whether interestless banking as practiced by Islamic banks is also a form of rationalizing away a religious mandate is debatable. On the other hand, the commands given in the Bible and the Qur'an seem to refer primarily at least to the practice of moneylending by those who profit off of misery; banking without interest (accepting for the moment the economic argument that Islamic banks operate on the basis of interest as well) is not really possible in a modern financial system. Islamic banking is, then, a similar answer to the revision of the Catholic Church's doctrine on usury: an attempt to reconcile the specific commands of religious texts and the needs of a far more sophisticated society with financial arrangements hardly conceived of at the time the religious documents were first recorded.
References
The Institute of Islamic Banking and Insurance (http://www.islamic-banking.com/)
The Islamic Bank of Britain (http://www.islamic-bank.com/)
"Islamic Banking", Mohamed Ariff (http://www.usc.edu/dept/MSA/economics/islamic_banking.html)
Dubai Islamic Bank (http://www.alislami.co.ae/)
"Is Islamic Banking Islamic?", Tarek El Diwany (http://www.irfi.org/articles/articles_301_350/is_islamic_banking_islamic.htm)