The Federal Deposit Insurance Corporation (FDIC) was created in 1933 during the Great Depression. It is not really a corporation as the name says. It is a government agency which guarantees people's deposits in banks. The amount of insurance has increased over time. Just a few years ago, it insured up to $100,000 per depositor per bank. That amount was increased to $250,000 during the fall of 2008.
The agency was created due to the large number of bank runs taking place in the 1930's. This caused the equivalent of monetary deflation, as bank runs reversed the fractional reserve process. This was one of the reasons that prices fell during the Great Depression, even though the central bank was engaging in monetary inflation (although not nearly enough according to Helicopter Ben Bernanke).
As a libertarian, I am completely opposed to the FDIC. Only the government could run such an agency. No private firm would find it profitable to insure banks in the current system of huge risk taking and massive fractional reserve lending. I don't think it is inconceivable that there could be insurance companies for depositors in a free market environment, but it would look a lot different than it looks today. There would be no central bank, far less (if any) fractional lending, and we can also assume that most banks would be far more careful with their lending standards.
With the current system we have now with fiat money, central banking, and government granted privileges, only the government can really run a deposit insurance agency. That is because only the government and the Fed have a legal monopoly over the money supply and can always create new money out of thin air if necessary. No private insurance company could do this.
I am a hard-core libertarian and I am in favor of abolishing most government programs and departments immediately. However, the FDIC might be one exception where I can envision major problems if the FDIC were abolished overnight. There would be massive bank runs immediately, and the last ones to arrive would be stiffed out of all of their money that was in a bank.
I hated the bank bailouts that happened in 2008. The politicians in DC really demagogued the issue. With that said, there may have been one relevant point when they were saying that we would see Armageddon if we didn't pass the bailouts. For some reason, most politicians were afraid to come out and say the obvious, which was that they feared massive bank runs. They threw all of these threats at the American people (including martial law), but I didn't hear many say that we needed to pass the bailouts so that you could still access your checking account and withdraw money out of the ATM on Monday morning.
The American people would have actually supported the bailouts if they were told this and thought it was their only choice.
However, it really wasn't the only choice. The government could have actually let the banks go into bankruptcy and just refunded depositors their money. The bank executives would have lost their jobs and their pensions. The government/ central bank would have had to use the printing press to do this, but it probably would have been less than what was handed out with TARP. Then the banks and all of their assets (for the ones that had to go into bankruptcy) could have been sold to the highest bidders. Any money collected by the government could have been immediately used to pay off the new bonds held by the Fed, thus reversing the previous monetary inflation.
This whole FDIC thing is a major problem that we need to get rid of. It is the ultimate moral hazard. Depositors do not really care what bank they use in terms of safety. Banks use excessive risk with this backstop. The only regulation on the banks is occurring by government. It should be done by the marketplace. Private insurance companies could play a big role in keeping banks sound.
I am not sure about how to end the FDIC in an orderly fashion. But as long as there is an FDIC, there will continue to be bailouts.