Islamic banking (or participant banking) is banking or banking activity that is consistent with the principles of Islamic law (Sharia) and its practical application through the development of Islamic economics. Sharia prohibits the fixed or floating payment or acceptance of specific interest or fees (known as Riba or usury) for loans of money. Investing in businesses that provide goods or services considered contrary to Islamic principles is also Haraam (forbidden). While these principles may have been applied to historical Islamic economies, it is only in the late 20th century that a number of Islamic banks were formed to apply these principles to private or semi-private commercial institutions within the Muslim community.

 
Riba

The word "Riba" means excess, increase or addition, which according to Shariah terminology, implies any excess compensation without due consideration (consideration does not include time value of money). The definition of riba in classical Islamic jurisprudence was "surplus value without counterpart", or "to ensure equivalency in real value", and that "numerical value was immaterial."

Applying interest was acceptable under some circumstances. Currencies that were based on guarantees by a government to honor the stated value (i.e. fiat currency) or based on other materials such as paper or base metals were allowed to have interest applied to them. When base metal currencies were first introduced in the Islamic world, the question of "paying a debt in a higher number of units of this fiat money being riba" was not relevant as the jurists only needed to be concerned with the real value of money (determined by weight only) rather than the numerical value. For example, it was acceptable for a loan of 1000 gold dinars to be paid back as 1050 dinars of equal aggregate weight (i.e., the value in terms of weight had to be same because all makes of coins did not carry exactly similar weight)....
Modern Islamic banking

Interest-free banking seems to be of very recent origin. The earliest references to the reorganisation of banking on the basis of profit sharing rather than interest are found in Anwar Qureshi (1946), Naiem Siddiqi (1948) and Mahmud Ahmad (1952) in the late forties, followed by a more elaborate exposition by Mawdudi in 1950. The writings of Muhammad Hamidullah 1944, 1955, 1957 and 1962 should be included in this category. They have all recognised the need for commercial banks and their perceived "necessary evil," have proposed a banking system based on the concept of Mudarabha - profit and loss sharing.

In the next two decades interest-free banking attracted more attention, partly because of the political interest it created in Pakistan and partly because of the emergence of young Muslim economists. Works specifically devoted to this subject began to appear in this period. The first such work is that of Muhammad Uzair (1955). Another set of works emerged in the late sixties and early seventies. Abdullah al-Araby (1967), Nejatullah Siddiqi (1961, 1969), al-Najjar (1971) and Baqir al-Sadr (1961, 1974) were the main contributors.

The early 1970s saw institutional involvement. The Conference of the Finance Ministers of the Islamic Countries held in Karachi in 1970, the Egyptian study in 1972, the First International Conference on Islamic Economics in Mecca in 1976, and the International Economic Conference in London in 1977 were the result of such involvement. The involvement of institutions and governments led to the application of theory to practice and resulted in the establishment of the first interest-free banks. The Islamic Development Bank, an inter-governmental bank established in 1975, was born of this process.

The first modern experiment with Islamic banking was undertaken in Egypt under cover without projecting an Islamic image—for fear of being seen as a manifestation of Islamic fundamentalism that was anathema to the political regime. The pioneering effort, led by Ahmad Elnaggar, took the form of a savings bank based on profit-sharing in the Egyptian town of Mit Ghamr in 1963. This experiment lasted until 1967 (Ready 1981), by which time there were nine such banks in country.
 
Sources:

        http://icif.in
        http://en.wikipedia.org/wiki/Islamic_banking