If bank bailouts don't happen, then due to fractional reserve banking they will become insolvent, and their depositors and counter-parties will also become insolvent. This means the "savings" of people like you will vanish into thin-air, like what happened in the great depression. Now, you may be wondering how all of this could happen with FDIC, however, FDIC only has roughly $45B in funding to cover $1T (or possibly much more) in at-risk deposits. The men behind the curtain have decided to bail out the banks before bailing out FDIC, as presumably this would be less catastrophic.
Of course, the real question that isn't being answered, at least by the media or other pundits, is how did we get into this mess in the first place? The answer, of course, is that it was inevitable given the system. We have a system of endless inflation and monetization of debt, coupled with fractional reserve banking, which is inherently unstable. No one typically complains during the boom phase, when money is cheap and prosperity in the form of ever-rising real estate prices seems limitless. But like every great party, the hangover can be brutal. It will take years to reconcile all of the mismanagement and malinvestment that has occurred due to the Federal Reserve, and it will be painful.