Bank Runs: Bottleneck at the Exits
A bank run is a customer stampede for the exits. Everybody takes out as much money as possible, and the institution bleeds deposits.
Banks have a limited amount of cash and limited resources to distribute the cash. As a result, a bank run clogs up the system with people desperate to get their money out.
Limited Reserves and Liquidity
Bank runs are problematic because the bank doesn't keep your money lying around in the vault. They might have 10% of customer assets in liquid form (called "reserve" money). Of that 10%, only a small portion is in physical cash such as bills and coins. The rest of the money is loaned out or invested elsewhere.
The system generally works well. Most customers don't need access to all of their cash - that's why it's in the bank. However, during a bank run everybody wants out, and they want all of their money in hand.
Bank runs can also happen on a national level (as opposed to a run on a single bank). If people think that the banking system in a given country is about to collapse, they'll pull money out as quickly as possible.
Bank Runs and Panic
Bank runs are a result of panic. Customers suspect that there's a problem, and nobody wants to be the last one out the door. As they see other customers heading for the exits, the sense of urgency grows. The process ultimately spirals out of control.
Are Bank Runs Necessary?
In general, bank runs are unnecessary. Most depositors in the United States will not lose money if their bank fails. In fact, they might not be inconvenienced in any meaningful way. The FDIC ensures that covered customers will get their money, within certain limits.
Those who are not fully covered by the FDIC are in a worse situation, and it may be worth their while to participate in a bank run. Likewise, a total collapse of the financial system might warrant a bank run, but what would your money be worth anyway?