All very good answers for the most part, but I have a definition for Good and Bad debt which is a little bit different from those mentioned here so far. The definitions come from Robert Kiyosaki in his book "Rich Dad Poor Dad", which I have applied to all my debts.
Good Debt - Good Debt is debt used to fund a money making asset, an asset which puts money into your pockets (or bank account) each month. In other words the income produced by that asset is more than all the expenses (including the interest repayments on the debt) associated with the asset.
Bad Debt - Bad Debt is debt used to fund both money losing assets and non-assets, where the interest repayments on the debt are more than any income (if any at all) produced by the goods or services the debt was used to purchase, so that you need to take money out of your pockets (or bank account) each month to sustain the debt.
Based on this definition a mortgage used to purchase the house you live in would be classed as bad debt. Why? Because you are making interest repayments on the mortgage and you have other expenses related to the house like rates and maintenance, but you have no income being produced by the house.
Even a mortgage on an investment (or rental) property where the rent is not enough to cover all the expenses is considered to be bad debt.
For the debt on an investment property to be considered as good debt, the rent would have to cover the full interest payments and all other expenses. In other words it would need to be a positively geared (or a cashflow positive) asset.
Why is this definition important in distinguishing between good and bad debt? Because it looks at the cashflow associated with the debt and not the profit. The main reason why most investors and businesses end up selling up or closing down is due to insufficient cashflow. It may be a profitable business, or the value of the property may have increased since you bought it, but if you don't have enough cash every month to pay the bills associated with the asset you will need to sell it.
If the asset produces enough cashflow to pay for all the expenses associated with the asset, then you don't have to fund the asset through other sources of income or savings. This is important in two ways. Firstly, if you are working and suddenly lose your job you don't have to worry about paying for the asset as it is more than paying for itself. Secondly, if you don't have to dig into your other source/s of income or savings to sustain the asset, then theoretically you can buy an unlimited number of similar type assets.
Just a note regarding the mortgage to buy a house you live in being classed as bad debt. Even though in this definition it is considered as bad debt, there are usually other factors which still can make this kind of debt worthwhile. Firstly, you have to live somehere, and the fact that you have to live somwhere means that if you did not buy the house you would probably be renting instead, and still be stuck with a similar monthly payment. Secondly, the house will still appreciate over the long term so in the end you will end up with an asset compared to nothing if you were renting.
Just another note to mention the definition provided by John Stern "...debt is a technology that allows borrower to bring forward their spending; it's a financial time machine...", that's a clever way to think of it, especially when it comes to good debt.