The Medici got rich by lending money without officially charging interest.
In the 1400s the Church banned usury, meaning any return on a loan beyond the principal. The Medici Bank, then Europe's largest, made its money anyway, through bills of exchange that turned currency conversion into a way to earn a profit. The money funded art, and put the family in charge of Florence.
Giovanni di Bicci de' Medici set up his bank in Florence in 1397. Its great early asset was the papacy: in 1427 the Rome branch held about 100,000 florins of papal deposits, while the whole bank's own capital was only around 25,000. Branches followed in Venice, Geneva, Bruges, London, Avignon, Milan and elsewhere. Each was a separate partnership in which the Medici held the controlling share, a structure historians compare to a modern holding company.
The Church held that whatever exceeds the principal is usury, so the bank could not simply pay interest on deposits and lend at a higher rate. Its main tool was the bill of exchange. A merchant paid florins to one branch and received a bill that another branch, say London, would pay out later in pounds at a rate fixed in advance. The time delay and the exchange rate hid a return, and because rates could move either way, there was enough risk that theologians did not count it as usury, except in 'dry exchange', where no money really moved.
The bank's most profitable years were from 1435 to 1455 under Cosimo de' Medici, who became the unofficial ruler of Florence. After his death in 1464 it declined. Its historian Raymond de Roover blamed distracted Medici heirs, weak oversight of branch managers, and ruinous loans to rulers: Edward IV of England, amid the Wars of the Roses, could not repay, and the London branch closed in 1478 with losses of more than 51,000 florins.
The end came in 1494, when the Medici were driven out of Florence, the central office was burned by a mob, and the remaining assets were seized for creditors. One popular claim is that the Medici invented double-entry bookkeeping. They used and refined it, but the earliest known double-entry ledger comes from a Florentine firm around 1299, a century before their bank existed.
Source: Wikipedia — Medici Bank · Text summarised from Wikipedia (CC BY-SA 4.0)