If you don't understand how an investment bank like Bear Stearns, which doesn't have depositors, can suffer a bank run, Felix Salmon does an excellent job of explaining:
They don't take retail deposits from individuals: there's no such thing as a Bear Stearns checking account, as far as I know. But that doesn't mean they can't suffer from a bank run. Except in this case it wasn't individuals withdrawing money, but hedge funds and other banks.
Bear had a large balance sheet, full of highly-rated bonds. If it ever needed cash, it could go to the repo markets and essentially borrow money against its own balance sheet: it would sell the bonds to a counterparty, and promise to buy them back at a slightly higher price the following day or the following week.
But then, last week, the repo window slammed shut for Bear. Other banks would no longer accept Bear Stearns as a counterparty, which meant that Bear couldn't use its balance sheet to raise cash.
Then, to make matters worse, the hedge funds all started deserting Bear as well. Bear has a large prime brokerage operation: it looks after hedge funds' assets, basically, and will lend them money against those assets as needed. But the hedge funds, worried that Bear didn't have the money to lend, started moving their assets elsewhere, and Bear's highly profitable prime-brokerage franchise started spiralling downwards.
Without the trust of other banks or hedge funds, Bear was toast. And the famously sharp-elbowed Bear was never much loved on Wall Street to begin with. Maybe it's true that the rest of the Street was simply waiting for an opportunity to get back at Bear for real and imagined slights, including the refusal to play ball in 1998. But I don't think the Fed was bearing any grudges; its job was simply to mop up in the aftermath.