The Federal Deposit Insurance Corporation is the government organization the insures most bank deposits up to $100,000 per person. This was created by FDR to help bring stability to the banking system and avoid a run on the bank during financially uncertain times.
In 1935 the insurance limit was $5,000. This grew over time and in 1980 it was raised to $100,000. In 2008 this limit was temporarily raised to $250,000 to help fend of some of the economic uncertainty of the times. This was done in an effort to bolster confidence in the banking industry and make people with large sums of money feel safer about their holdings to help prevent a run on the banks from high net-worth individuals.
The insurance limit operates on a per person, per institution basis. If you exceed the limit at one institution, you can move some of the money to another bank and have a $100,000 (temporarily $250,000) limit there as well.
Banks pay into FDIC insurance. FDIC has the authority to change bank management or force a sale if the bank's capitalization rates to to certain levels.
Federal Deposit Insurance serves an important cornerstone in the economic stability of the US banking system. It basically allows banks to operate in a way that makes them fully backed by the US government without creating a government owned bank.