In this very short video clip, Carl Denninger quickly explains this problem on RT TV and RealEcon TV.
In laymen's terms, here's a fleshed-out explanation of what Carl is saying:
European banks (and to some extent US banks) have a huge amount of assets, but those assets are mostly state debt (i.e. bonds purchased by the banks from European governments -- in other words loans they have made to those governments.)
However, the banks have no collateral for these loans they've made. (Example: when you take out a mortgage on the house you are buying, the bank has your house as collateral; when you buy a new car, the bank has the legal right to seize "your' car if you should stop making payments on the car. Its collateral is "your' house and/or "your' car.)
In other words, if something goes wrong and these governments become unable to buy back the bonds (plus interest) that they sold to these big banks, then the customers of the banks are in trouble. Why? Because the bank loaned out 40 or 50 dollars to these governments for every dollar of depositor money it had in its possession. That's called "leverage': if you're a bank you can, by law, create some specified amount of money, say $50, out of thin air, with which to buy bonds (or whatever), for every dollar your customers have deposited on your bank. But if the government whose bonds you have purchased cannot redeem those bonds (i.e. pay back the loan), then (at least in Europe) your depositors and stockholders are screwed.
Another way to think of "leverage' is that when a bank has an asset, such as customer deposits, it then loans those assets out more than once, perhaps many times more than once. Historically, leverage of more than five- or six-times deposits were considered dangerous. In the late '90s JP Morgan was leveraged 10-12 times its deposits. Nowadays, leverages of 40-50 times deposits are tolerated by our regulatory agencies.
American depositors are protected, up to $250,000 per account, by the FDIC. If Spain defaults and BofA goes belly-up, the bank's shareholders get screwed, but the bank's customers' deposits are still safe -- as long as the FDIC is solvent! BofA's remaining assets, including the deposits and loan money owed to it, will get divvied up among its creditors, and life goes on. (But if the FDIC isn't able to cover the deposits, we've got much, much bigger problems than a failed bank.)
Of particular concern is that a lot of these big banks are invested in each other. This means that the actual value of CitiCorp stock could take a dive if BofA's stock dives. Depending on how much flexibility there is within the system, the shock of a couple of mega-banks failing could conceivably be distributed throughout the rest of the financial system. Otherwise, the entire system could fall apart. Problem is, no one knows for sure how much flexibility there really is within America's overall financial system.