A bank run (also known as a run on the bank) occurs when a large number of bank customers withdraw their deposits because they believe the bank is, or might become, insolvent.
As a bank run progresses, it generates its own momentum in a kind of self-fulfilling prophecy: as more people withdraw their deposits, the likelihood of default increases, and this encourages further withdrawals. This can destabilize the bank to the point where it faces bankruptcy.
A banking panic or bank panic is a financial crisis that occurs when many banks suffer runs at the same time. A systemic banking crisis is one where all or almost all of the banking capital in a country is wiped out. The resulting chain of bankruptcies can cause a long economic recession. One of the direct causes of the Great Depression was a series of bank runs. The cost of cleaning up a systemic banking crisis can be huge, with fiscal costs averaging 13% of GDP and economic output losses averaging 20% of GDP for important crises from 1970 to 2007.
A bank run only happens when you have fractional reserve lending, otherwise there would always be enough money to cover the demands for withdraw. These bank runs were often used to convince people that we needed a central bank or so called lender of last resort. The underlying problem is always fractional reserve lending. So long as this practice is allowed to continue there will be regular and wide spread bank failures. A central bank just covers up these failures by printing more money.