[A:]Thanks for your question! First a definition.
Definition of a Bank RunThe Economics Glossary gives the following definition for a bank run:
"A bank run takes place when the customers of a bank fear that the bank will become insolvent. Customers rush to the bank to take out their money as quickly as possible to avoid losing it. Federal Deposit Insurance has ended the phenomenon of bank runs."
Bank Runs and how Banks WorkWhen you deposit money in a bank, you will generally make that deposit into a demand deposit account such as a checking account. With a demand deposit account you have the right to take your money out of the account on demand, that is at any time. However banks do not keep all the money in demand deposit accounts stored in a vault. Instead they take that money and give it out in the form of loans or other invest it in other interest paying assets. Banks are required by law to have a minimum level of deposits on hand, known as a reserve requirement. As detailed on the website of the New York Federal Reserve the reserve requirements are quite low, generally in the range of 10%. So at any given time a bank can only pay out a small fractions of the deposits of its depositors.
The system of demand deposits works quite well unless a large number of people demand to take their money out of the bank at the same time. This generally does not happen, unless people think their money is not safe in the bank. So a bank run typically occurs when the depositors of a bank believe that a bank may go insolvent and if they do not take their money out right away they may lose that money forever.