You must mean the current debt ceiling debacle.
The meaning of it is: US government is constantly borrowing money (by issuing treasury bonds) and constantly repaying some of the bonds that come to maturity, and also has other obligations it has to meet by law all the time - such as Social Security checks, bonds interest, federal employees' salaries and pensions, etc.
By law, total amount of money that can be borrowed at the same time is capped.
That means, there can be situation where the government needs to borrow money to pay, say, interest on existing bonds, but can not, since the limit is reached.
Such situation is called a default, since the government promised to pay the interest, but is unable to do so.
That does not mean the government has no money at all and will completely collapse or couldn't raise money on the market if it were permitted by law to do so (currently, the market is completely willing to buy the debt issued by US government, and with interest that is not very high, though of course that may change).
It also does not mean the economy ceases to function, dollars cease to have value or
banks instantly go bankrupt.
But if the government breaks its promises to investors, it has various consequences such as raising the costs of borrowing in the future.
Breaking promises to other people - like Social Security recipients - would also look bad and probably hurt many of them.
Going back to your bank account, most probably nothing would happen to the money you store there.
Even if the bank had invested 100% of the money in US treasury bonds (which doesn't really happen)
they still can be sold on the open market, even if with some discount in the event of credit rating downgrade, so most probably your account would not be affected.
As stated in another answer, even if the fallout of all these calamities causes a bank to fail, there's FDIC and if your money is under insured maximums you'll be getting your money back.
But if your bank is one of the big ones, nothing of the sort would happen anyway - as we have seen in the past years, government would do practically anything to not allow any big bank failures.
money.stackexchange.com/users/2049/StasM
In principle, a default will have no effect on your bank account.
But if the US's credit rating is downgraded, the knock-on effects might cause some more bank failures, and if the debt ceiling is still in place then the FDIC insurance might not be able to pay out immediately.
money.stackexchange.com/users/2325/Mike Scott
I have been through default in Ukraine august 1998.
That was a real nightmare.
The financial system stopped working properly for 1 month, about 30% of businesses went bankrupt because of chain effect, significant inflation and devaluation of currency.
So, it is better to be prepared, because this type of processes result in unpredictable situation.
money.stackexchange.com/users/4214/columbus
If you are actually referring to all the political rhetoric and posturing over the debt ceiling issue.
That's a long ways from the US actually defaulting on paying debts.
A lot of government offices might shut down, but I expect anyone holding US debt to be paid off.
(they have the printing presses after all)
If that's what you are referring to, based on the LAST time that the governement had to shut down because they didn't raise the debt ceiling, it won't be a big deal.
Last time, no debt was defaulted on, a bunch of the less essential government offices shut down for a few days, and the stock market did a collective 'meh' over the whole thing.
It was basically a non event.
I've no reason to expect it will be different this time.
(btw, where were all these republican budget cutters hiding when 10 years ago they started with a nearly balanced budget, and ended up blowing up the national debt by about 80% in 8 years time?
(from roughly $6B to $11B)
I wish they'd been screaming about the debt as much then as they are now.
Not that there isn't ample blame to go around, and both sides have not been spending in ways that make a drunken sailor look like the paragon of a fiscal conservative, but to hear nearly any of them tell it, their party had nothing to do with taking us from a balanced budget to the highest burn rate ever while they were in control (with a giant financial crisis through in as pure 'bonus')
money.stackexchange.com/users/3924/Chuck van der Linden
Government default doesn't mean that all US money is immediately worthless.
First, the bondholders will get stiffed.
Following that, interest rates will shoot up (because the US is a bad credit risk at this point) and the government will monetize its ongoing expenses -- i.e., fire up the printing presses.
If you're concerned about not having access to your money, start pulling out a little extra when you get cash at an ATM.
Build it up over time until you have enough currency to weather through
whatever emergency you envision with your bank account.
money.stackexchange.com/users/414/mbhunter
Will I still be able to withdraw from my bank account if the U.S. defaults?
If the U.S. defaults on its debt, what will happen to my bank money?
money.stackexchange.com
<united-states><bank-account><sovereign-default>
money.stackexchange.com/questions/9807
money.stackexchange.com/users/4242/user541686
In principle, a default will have no effect on your bank. The knock-on effect of a government default might be banks failing, but even if your bank were to default, your money is still insured by the FDIC.  In principle, a default will have no effect on your bank account or your money stored there. Government default doesn't mean that all US money is immediately worthless.  Your money is also insured with FDIC, but if the US's credit rating is downgraded, the knock-on effects might cause more bank failures, and if the debt ceiling is still in place then the FDIC insurance might not be able to pay out immediately. If you're concerned about not having access to your money, start cashing out a little extra whenever you get cash at an ATM.  
In principle, a default will have no effect on your bank account or your money stored there. Government default doesn't mean that all US money is immediately worthless.   Even if fallout a bank to fail, there's FDIC and if your money is insured so you'll be getting your money back. But if the US's credit rating is downgraded, the knock-on effects might cause more bank failures, and if the debt ceiling is still in place then the FDIC insurance might not be able to pay out immediately.  If you're concerned about not having access to your money, start pulling out a little extra when you get cash at an ATM.  