A “bank run” is a “run on a/the bank,” when panicked bank customers demand their money before a seeming bank collapse. Deposit insurance protections in the United States (put in place after the Great Depression) have relieved some of the fears of bank runs.
 
The term “run on the bank” has been cited in print since at least 1735.
 
 
Wikipedia: Bank run
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 (or positive feedback): 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. Much of the Great Depression’s economic damage was caused directly by 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.
 
Several techniques can help to prevent bank runs. They include temporary suspension of withdrawals, the organization of central banks that act as a lender of last resort, the protection of deposit insurance systems such as the U.S. Federal Deposit Insurance Corporation, and governmental bank regulation. These techniques do not always work: for example, even with deposit insurance, depositors may still be motivated by beliefs they may lack immediate access to deposits during a bank reorganization.
 
BusinessDictionary.com
run on the bank
A surge of individuals withdrawing funds from a financial institution due to fear that the entity will become insolvent in the near future. This typically results in long lines of people waiting outside of the institution to withdraw funds or transfer them to another institution
 
The Free Dictionary
Bank Run
An event in which many account holders at a bank withdraw all of their funds at the same time because they do not believe the bank is solvent. Ironically, the pressure of a bank run itself can cause the bank to become insolvent. In the United States, bank runs were fairly common before the creation of the FDIC, which insures bank deposits up to a certain amount.
 
Google Books
October 1735, The London Magazine, or, Gentleman’s Monthly Intelligence, pg. 556, col. 1:
several eminent Goldsmiths had stopp’d, which caus’d so great a Run on the Bank, that it is generally though the Bank must have stopp’d, had it not been for the Subscription they took for the Circulation.
 
7 January 1746, Pennsylavnia Gaette (Philadelphia, PA), pg. 2:
Wednesday last there was a great Run on the Bank, supposed to be by Jacobites and others, whose Relations are now in actual Rebellion;...
 
7-14 May 1771, Essex (MA) Gazette, pg. 168:
So great has been the run on the Bank, since the commitment ofthe Lord-Mayor and Mr. Alderman Oliver to the Tower, that betts were laid on Saturday last at New lloyd’s, to pay 30 guineas to receive 100, if Bank notes were not discounted before the expiration of one month.
 
27 June 1787, New York (NY) Independent Journal, pg. 2:
Paris, April 11. At this moment the utmost confusion reigns here, owing to a general distrust and want of current cash. The bankers offer 12 per cent. for money to support their credit; and, to add to the calamity, a run on the bank (the Caisse d’Escompte) has already begun. 
 
22 February 1817, Exile (New York, NY), pg. 2:
RUN ON THE BANKS
The expected issue of specie, which is to relieve the public from the miserable paper now in circulation, has caused a very considerable run on the banks, as a foolish idea has been taken up that the banks will not be obliged to discharge the claims made on them for the two-penny notes after a certain period.
 
3 March 1826, Baltimore (MD) Gazette and Daily Advertiser, pg. 2:
A Bank run.—The banks at Nantucket, which lately stopped payment in consequence of some misunderstanding with the Boston banks, and closed their doors for the purpose of arranging their affairs, opened their vaults some days since.