What exactly is a bank failure?

a little history before I answer your question. In the early 1400's, there were no banks. There were goldsmiths though. Due to the nature of their work, Goldsmiths had a lot of gold on hand and the security forces to protect them. So, for protection against bandits and raiders, people started to deposit their gold and valuables with the Goldsmiths for protection. Of course the Goldsmiths would charge fees for this service. As commercial trade opened accross the the globe, it became increasinly painstaking to travel back to where one had there gold statched just to make a withdrawl. To remedy the problem, Goldsmiths began to allow depositors to make paper gurantees against deposited gold and use those paper notes as tender to trade. Think of checks here. Goldsmiths started to realize that people would make deposits and not even touch them for a while. This gave Goldsmiths the idea of loaning the gold to people for a profit since it was just sitting there and they could cover any loaned amounts from their own gold reserves. This system caught on in a time of global trade and the first banking institutions were born.
Today's banks are more that a deposit depo. Banks make loans, mortagages, as well as offer investment vehicles like CD's and Money Market Funds. When banks loan money, where do you think they get that money from? They get that money from customer deposits just like the Goldsmiths did in the beginning. It is for this reason banks are required to keep reserve requirements on hand at any time. Currently, Bank reserve requirements are 20% of deposits. This means they can loan 80% of your deposit at anytime. Banks do this because most people don't use their whole bank account in one day; they let those funds sit there for how ever long. Even if you withdraw all of your funds tomorrow the banks can cover that withdraw with new deposits and repaid loans. But what if the bank can't cover all the withdrawls? If the bank can't cover with cash the amounts customers have on the books, the bank fails. This can happen for many reasons. Loans are defaulted on, People make a run on the bank withdrawing all of their deposits at once, or they get robbed.

FDIC is insurance offered by the Federal Reserve to banks who keep there required bank reserves in their mints. FDIC was enacted after the great depression to ensure people that their deposits are safe. You see during the Great Depression the banks failed. The banks didn't have enough cash to cover the deposit amounts on the books and when people caught wind of this they started to withdrawl their money in masses therefore casusing banks to go insolvent. Only cash deposits and safety deposit boxes are FDIC insured. FDIC only insures up to $250,000 dollars. FDIC is paid by banks out of "bank money" fees, interest, and dues charged by the bank for their services.