Medieval banking developed from the activities of money-changers. By the end of the C12th Templars were serving as, in effect, bankers to the kings of France and England, and transfered money from papal tax collectors to Rome.
Banking in the C13th was a simple operation, and written instructions to a banker developed slowly, however this was not the case in Italy, as it was simple to communicate in those Italian cities which became known for their banking services, Piacenza, Lucca or Florence.
Italian merchant-bankers appeared in London as early as 1224. But by the end of the C13th banking activity had shifted to Florence. Many banks were family partnerships, the largest the Bardi, Peruzzi and Acciaiuoli.
The medici bank was a group of partnerships which became famous in the later middle ages. The Medici became used for everything, including normal banking, and transmittion of funds from the Papal taxation to Rome. The Medici had baks throughout europe in Bruges, London, and Venice. Expansion came when many of its smaller rivals became bankrupt, due to the unbalanced nature of trade in this period. It was also the development of the bill of exchange which led to Italian dominance of banking in the C14th, due to the large profits made on transactions. These transactions of course usually took place in the Italian cities and so led to their dominance.
It was also the use of book-keeping which made Italian benking rise above the rest. By the C14th they had learned to keep double-entry book-keeping in Genoa and even earlier in Tuscany. It was used throughout Italy and the Italian banking houses of western Europe but apart from this it was not widespread. i.e. It seemed exclusive to the Italian bankers.
The shortage of currency in trade in medieval period, aslo gave rise to the Italian banks as more transactions were being done without the use of currency.