A bank failure is the closing of a bank by a federal or state banking regulatory agency. Typically, a bank is closed when it becomes critically undercapitalized or is unable to meet its obligations to depositors and others.
The FDIC is then appointed receiver (by the regulatory agency of the bank in question) and assume the tasks of:
(a) Disposing of the failed bank's assets in a manner that maximizes their value, and
(b) Settling the failed bank's debts, including claims for deposits in excess of the insured limit.
A bank failure does not change your obligation as a borrower to make payments and comply with the terms of your loan.