A Bank run is a panic response which occurs when a large number of people in a short time take their savings out of a bank, which they fear is financially unsound and about to collapse. This is much more likely to occur if the bank exists within a gold standard (or other specie-backed) system, where bank notes are redeemable for gold.
A bank run begins when the public begins to suspect that a bank may become insolvent. As a result, individuals begin to withdraw their savings. This action can destabilise the bank to the point where it may in fact become insolvent: no bank has enough reserves on hand to cope with everyone taking their savings out at once. It is this gap which banks exploit to earn interest. The increasing number of people removing their savings can destroy the ability of the bank to do business, thus forcing it into insolvency. It is therefore people trying to protect their savings that precipitates the ultimate collapse of the bank.
A bank run has much in common with the reflexive processes described by George Soros, amongst others. Another example of a reflexive process is economic bubble.