First, there are a number of different measures of National Debt. The full federal debt is at $5.6 trillion. If the intragovernment debt is included (money borrowed by the government from its own Reserve or its own agencies), the figure is about $9 trillion. If you include all of the promised social security and medicare/caid payments, our debt is $59 trillion. The last measure is misleading, because it is taking into account all of the unpaid social security and public health benefits for everyone in the country. Under this assumption the United States is constantly in massive debt. It seems the $9 trillion figure is used the most and is figured into the data.
Valkyrie is also correct, slick Willy paid off the Debt during his presidency. This was done largely by downsizing the military and raising taxes.
National debt has an inflationary effect on currency, because the government has to add money to the economy and each dollar becomes worth less. Having too large of a debt will cause massive inflation to happen. This happened to Germany after WWI because of the reparations they owed. When it started to cost billions and billions of Deutsche Marks for a loaf of bread, people stopped using money and the economy fell (this allowed the Nazi party to come in and take over). Having a national debt is beneficial as well, because it prevents the value of the dollar from depreciating too much and the US having the problem of too little money in the economy.
About 25% of the national debt is owed to foreign governments. Of this, 47% is owed to Japanese and Chinese lenders. Overall, 44% of it is owned by foreigners, and 66% of this is owed to the central banks of foreign countries. China has actually helped us by holding the national debt in our own dollars and not in their currency, allowing out inflation to stay very low.
According to the article, (straight from the government statisticians),
"The US budget deficit has been declining for 3 years and is expected to be again in surplus by 2012. When the U.S. Government has a surplus, it may pay down its outstanding debt. It does this by paying back the principal of the outstanding bonds redeemed for payment while not issuing new bonds. The U.S. Government could also purchase its own outstanding securities on the open market if it was searching for a way to use a surplus to reduce outstanding debt that was not due for redemption in a given year."
So, if nothing changes by 2012, we will have a yearly surplus in our budget and can begin to pay the debt off.
Our spending in Iraq has contributed greatly to raising the debt. Our unwillingness to move production facilities to the United States (and thus import more goods than we export) has contributed greatly. The Bush tax cuts have contributed to the issue (although the economic stimulus that they provided could have raised more in taxes than they cost, I'm not sure anyone has calculated that).
Here is a pretty detailed article:
http://en.wikipedia.org/wiki/United_States_public_debt
Regardless, each American owes about $27,000 to ourselves as well as foreign governments. The only ways to lower that figure are to cut spending, raise taxes, and/or start to export more than we import.
As for Hillary's plan, it seems like an attempt to gain the appeal of young voters that don't understand how the national debt works.