Washington had so much fun with its artificial crisis last month (the fiscal cliff) that it's doing it again next month (the debt ceiling). President Obama has said he will not negotiate over raising the debt ceiling ("While I will negotiate over many things, I will not have another debate with this Congress about whether or not they should pay the bills they have already racked up") and Republicans claim they are willing to kill the hostage (Pennsylvania Senator Pat Toomey: "we Republicans need to be willing to tolerate a temporary, partial government shutdown in order to achieve spending cuts and entitlement reforms.").
So, what is the debt ceiling? The debt ceiling refers to the legal limit the U.S. government may be in debt. Any increase in the debt ceiling has to be approved by Congress. If the debt ceiling is not increased in time (which happened in 2011), the government defaults on its loans, because the government is legally obligated to pay the bills it has committed to. Because Congress can let the government shut down and default by not increasing the debt ceiling, the debt ceiling is routinely used to extract a few compromises, but it was only in 2011 that Republicans actually let the government default.
President Obama does have a bizarre potential workaround: the trillion dollar coin. The treasury has the right to mint coins of any value, mostly as collector's items, but there is nothing stopping the treasury from minting a trillion dollar coin, and using that coin to keep the U.S. solvent. The trillion dollar coin option is unlikely, mostly because it would make for terrible publicity and political spin, but you'd think the same would be true of voluntarily allowing the government to default.