The debt ceiling, or statutory debt limit, limits the amount of public debt that can be outstanding. Currently it is ~14.2 trillion. They have to increase it so that the U.S. Treasury can legally issue more debt, to go beyond 14.2 trillion. This is a consequence of the annual deficits, where the annual expenditures (entitlements, wars, departments, etc) exceed the tax receipts (individual income tax, corporate tax, etc). Compound interest does play some part, but it is relatively minor. The major problem is that right now the the politicians in Washington spend $3.8 trillion, but only receive $2.4 trillion in taxes. This is a shortfall of $1.4 trillion that needs to be "covered" somehow, and it is done through issuance of debt.
Deficit = how much further we go into debt (usually an annual number), $1.4 trillion right now
Debt = how much we owe in total, $14.2 trillion right now
(Rand stumbles on these in his speeches)
So simple example, you have a credit card with a $14,200 limit that is maxed out, no remaining balance. You make $5,000 this month, but will spend $6,000. ($1,000 deficit, $14,200 debt) You would go to your credit card company and ask them to raise the limit to $15,200 so that you can cover your bills for the month. If the card company says no and will not raise the credit card limit, then all of a sudden you're screwed and you have to figure out your priorities and how to spend only $5000 this month.
Congress raises their "credit card" limit by raising the debt ceiling. In the hypothetical situation where a Senator filibusters the legislation to increase the debt limit, then all of a sudden Congress needs to figure out how to make $2.4 trillion in income (tax receipts) pay for $3.8 trillion in expenditures - they can't and hard decisions on spending less need to be made. HOWEVER, what the MSNBC people are doing is fearmongering. The U.S. honors its debts, so it would pay the interest on the $14.2 trillion FIRST, which amounts to ~$400 billion right now. That leaves $2 trillion to pay for $3.4 trillion in welfare/warfare and then some really, really, really hard decisions need to be made by our masters. Would they stop funding Afghanistan and Iraq? Would they rather pay less to Social Security/Medicare/Medicaid recipients? Both? Something else?
The only "Armageddon" that would occur is the politicians having to tell voters that they actually had to balance the budget, that we don't have money for everything, and a lot of groups are on the losing end. Social Security/Medicare/Medicaid/Defense MUST be cut to balance the budget, which would guarantee that most politicians would get killed in reelection. The amazing thing is that "one-time" stimulus from the Bush/Obama administrations have become a structural annual expense that no one can undo anymore. Going back to "only" the $300 billion deficits under Bush 2 cannot be done according to Boehner and Cantor. (They are shooting for 2008 expenditures, which still leaves us in massive deficits) The reality is that the budget will never be balanced until someone, or some unfortunate event, forces them to make the hard choices.