What was the impact of the introduction of money on ancient Babylon and its environs?

Economics as we know it.

Before the introduction of money, most of what we think of as economics was impossible. Goods were produced primarily by the families who used them, and when goods were exchanged between families it was based on individual relationships and social norms. People would give to the needy, support distant relatives, make deals with others in the village. They did not "buy" or "sell" in the modern sense, and in fact even bartering in the sense of a direct exchange of goods was quite rare. It would be much more common to give someone something with the tacit expectation of a favor, or to call in some other favor to get some goods. Economics was essentially based on promises and relationships; in order to sustain that, it was necessary to do almost all exchanges among people who knew each other well and could trust one another.

The invention of money allowed people to begin buying from and selling to people they barely knew; by exchanging a piece of gold or silver for the good, both parties could be ensured that they were getting a fair exchange. Money's central function was to obviate the need for trust in economic transactions. You did need to trust the government to maintain the value of the money---even under a classical gold standard it is possible to debase coins, and this happened quite often---but you didn't need to trust the individual vendor with whom you were trading.

This in turn allowed trade to occur over much longer distances, and encouraged more specialization and division of labor. These two effects in turn greatly increased the efficiency of production, allowing a larger quantity of goods to be sold, larger populations to be sustained and standards of living to rise. By separating economic transactions from personal relationships, money allowed individuals and even whole nations to specialize in the production of particular goods and thereby achieve greater efficiency.

In Babylon in particular, the establishment of money was associated with the establishment of complex legal codes and government institutions. Hammurabi's Code, one of the oldest and best-preserved legal codes, was created during this time, and it includes  sections on taxes, fines, debt, and even interest rates. We're not sure which came first---laws or money---but the two are clearly highly interdependent. The invention of money marked a major epoch in the rise of Mesopotamian civilization. Remember, Mesopotamia stood for thousands of years before it fell to Alexander the Great in the 4th century BC; part of how it was able to do that was by being one of the first civilizations in the world to have a well-developed monetary system.

Of course, there was a downside: The banks and governments who controlled the money system gained an enormous amount of power they previously did not have, because the money they controlled now had the power to lay claim on almost any sort of goods or services. While total wealth increased dramatically, so did the inequality of its distribution. In a time where most people could not read or do arithmetic, even a basic level of financial literacy provided a huge advantage in society.  Even today, people who understand and control money are a lot more powerful than people who don't!