The Federal Reserve Act of 1913 established the Federal Reserve System as the central bank of the United States to provide the nation with a safer, more flexible, and more stable monetary and financial system. The law sets out the purposes, structure, and functions of the System as well as outlines aspects of its operations and accountability. Congress has the power to amend the Federal Reserve Act, which it has done several times over the years. The complete act, as amended, is provided here by section.

The Federal Reserve System is the central bank of the United States.  It performs five general functions to promote the effective operation of the U.S. economy and, more generally, the public interest.

The Federal Reserve

  • conducts the nation’s monetary policy to promote maximum employment, stable prices, and moderate long-term interest rates in the U.S. economy;
  • promotes the stability of the financial system and seeks to minimize and contain systemic risks through active monitoring and engagement in the U.S. and abroad;
  • promotes the safety and soundness of individual financial institutions and monitors their impact on the financial system as a whole;
  • fosters payment and settlement system safety and efficiency through services to the banking industry and the U.S. government that facilitate U.S.-dollar transactions and payments;
  • promotes consumer protection and community development through consumer-focused supervision and examination, research and analysis of emerging consumer issues and trends, community economic development activities, and the administration of consumer laws and regulations.

Many developed countries have their own central banks, the largest of which are the Bank of Japan, the European Central Bank as well as the Bank of England, The Bank of Canada, the Reserve Bank of Australia and the Swiss National Bank.  Here are links to learn more about them:
https://www.federalreserve.gov/
https://www.snb.ch/en/
http://www.bankofcanada.ca/
http://www.bankofengland.co.uk/Pages/home.aspx
https://www.ecb.europa.eu/home/html/index.en.html