The 12 regional Reserve Banks act as the service division of the Federal Reserve -- they carry out the monetary policy set by the Federal Reserve Board and regulate and supervise financial institutions. The agency that charters the bank is also responsible for conducting on-site examinations to make sure the bank is complying with banking laws. In addition to this supervision, your money is also protected by insurance.
That "FDIC" logo you see as you walk in the door means that you hold insurance on your deposits. Depositors are typically protected for up to $100,000.
Deposit insurance came about because of rumors of banking trouble that lead to panics and everyone running to the bank to withdraw all of their money. It didn't take much to make people uneasy about the security of their money in the bank. If they heard of the slightest hint of trouble, they ran to the bank to withdraw. This lead to the failure of many banks and huge losses of savings for many people. This roller coaster of personal finance lasted for many years and throughout the Great Depression of the 1930s. Finally, in 1934, Congress established the Federal Deposit Insurance Corporation (FDIC), which initially provided deposit insurance coverage of $2,500 per depositor. This greatly improved the security of banks and reduced the number of bank failures by almost 4,000 from 1933 to 1934.
Public confidence in the banking system has improved tremendously since the FDIC was established. The trust that depositors need in order to make the system work is maintained, and the economy keeps humming.
Banks also carry private banking insurance -- specially designed private coverage to protect deposits in the case of burglaries, robberies, vandalism, etc.