Basically, here's how this works.
In any normal country, once you pass a budget, then the treasury will pay for it. You only have to pay for something once. This is, of course, intuitive, proper, right, and logical- and so the US has to do something entirely different.
Instead of Congress putting a budget through and then just having that stuff be paid for (because, you know, it authorized all that shit already), it has to authorize such payment twice- once at the budget level and once with the debt ceiling. The debt ceiling was, until 2011, a mere formality. It was an entirely pointless piece of legislation (that we don't constitutionally have to have around, even!) that puts a cap on how much money the federal government can borrow. Once you've done borrowing up to that limit, you have to raise it to borrow more. For most of its history, the debt ceiling has been regarded by both sides as a necessary and sacred legal fiction, too holy to be debated at all.
Then, in 2011, the newly sworn in GOP-led Congress, with a bright new crop of Tea Party nuts ready to take action, found out that you could use the debt ceiling as a bargaining chip to try and get your unrelated domestic agenda enacted...because, despite the fact that this is not even a necessary fiction, once the ceiling has been hit, if the government goes for any length of time after that point, it will default on its debt. I don't know how much of its debt will be defaulted on, but this is an extremely bad thing. China holds trillions of dollars of our debt, which has historically been the safest investment in the world, bar none. Wars are waged, nations have fallen and risen, depressions and bubbles and booms and busts have come and gone, but the US Treasury has been selling its bonds for more than two centuries, and it has not been a minute late on a single one of them.
Naturally, then, the safest debt in the world sold by the most powerful nation in the world is a key cornerstone of the modern world economy. For that reason, if the US defaults on so much as a dime of its Treasury bonds, the entire world's financial markets are going to panic in a way that is likely to make 2008 look like a reasoned and deliberate response to a small error. We have no idea how bad it would get, but it would get bad. Very bad. All the more so when you consider that the GOP's demands in return for not causing the historical mire of the century are nothing less than undoing just about every major piece of legislation President Obama has enacted while in office.
Now, there are a few ways around this. For example, if the GOP refuses to budge come the 17th, Obama could instruct the Fed to mint a platinum coin with a face value of $1 trillion and place it in the US Treasury. This looks like running to the printing press to pay your bills, and rightly so- it is a mere accounting trick, and for that reason cannot be done more than once or twice. By keeping it in the Treasury, for now, it won't cause inflation, because it isn't out in the market being traded around, but that, again, is a legal fiction. It's a one-time method to keep Congress from destroying or severely crippling the modern economic world order. In the end, however, there is only one way around this, which is to vote the lunatics out of Congress. And unfortunately, without a real crisis, such as that which default would cause, this may not happen.