A bank run is what happens when a large number of a bank's customers decide to withdraw their deposits because they fear the bank will imminently go bust and hence be unable to repay them. This is especially acute in our era of fractional-reserve banking, where any given bank has much less money stored at any one time than it owes in deposits. The whole business model of a modern bank relies on customers believing that a bank run won't happen; when it does, the bank becomes insolvent, and those who withdraw their money first win.
Bank runs nowadays are just as likely to occur electronically and hence relatively invisibly as they are to involve thousands of people smashing in the windows of their local branch (we still do that in our era, but for different reasons). And, what with globalization and all that, bank runs can be trans-national and as much about business banking as retail banking: Ireland's banks have suffered a horrible capital drain lately as overseas businesses take their money back. Furthermore, the standard government policy to prevent bank runs - deposit guarantees, where the government promises to compensate lost deposits up to a certain amount - are sometimes only available to domestic customers.
If enough bank runs happen at once, you've got yourself a financial crisis. In economies that run mostly on debt - like many western ones - this is a big problem. When everyone loses confidence and starts calling in ther debts, the whole system can collapse very quickly, as the total amount of money owed across the economy vastly outstrips the amount of capital available to pay it back all at once. The confidence to assume new debts and to trust again returns slowly. We're nowhere near having it yet.