By far the most important role that the Fed plays in the US economy is its role as the maker of monetary policy. The Fed plays many other roles that are important, such as its role as the lender of last resort. However, no role is as important as its role in monetary policy.
Monetary policy has a huge impact on the US economy. For example, faulty monetary policy is often cited as one reason why the Great Depression was as bad as it was. By changing the money supply, the Fed has the ability to do a great deal to speed the economy up or to slow it down. It can do things like making it harder or easier for companies and individuals to borrow money. These sorts of actions are immensely important to our economy.
Thus, the Fed's most important role is as the maker of monetary policy.
Monetary policy is the regulation of the amount of money that is in the money supply. By manipulating the money supply, the Fed can try to keep the economy going in the right direction. If the economy is overheating, the Fed does things like raising interest rates and selling government securities. When the economy is stumbling as it is now, the Fed lowers interest rates and sells government securities as a way of creating more money.
The Fed's major role in the economy is to be an honest (not politically motivated) manager of monetary policy.