What impact did the Great Depression have on shaping the twentieth century?
It gave rise to sweeping changes in our banking system, stock exchange and many social program.
When the stock market crashed and so many banks failed it was obvious to all that sweeping changes needed to be made. Most of our banking regulations find their genesis in the reorganization of the national banking scheme. For example, our country went off the gold standard and the Federal Deposit Insurance Corporation came into being.
Wall Street became much more regulated. Prior to the crash people bought large amounts of stock on "margin." That meant a person could buy a million dollars worth of stock by putting up only $100,000. The balance owed was held by banks. This is no longer legal. In October 1929 the value of many stocks began to decline. As stocks lost their value the banks put a call on the money owed them by individuals who had purchased those stocks. The debtors in turn tried to sell off their stocks as the price per share was in free fall and in the end a share of stock was valued at a small percentage of its original face value. People and banks were bankrupted literally over night.
To get people back to work President Roosevelt devised a number of federally funded programs to get people working. Chief among them was the Works Progress Administration (WPA). Roosevelt declared that the nation's highways needed rebuilding along with its bridges. He also brought into being the Tennessee Valley Authority (TVA) which still exists today, the Civil Conservation Corps, and social security.
Much of what FDR started still exists today in one form or another. But it all moved American, albeit slowly, out of the depression. World War 2 was what finished the job.