The negative equity definition is when the the value of a loan is more than the value of the assets used to collateralize the loan. In addition, negative equity loans usually occurs because the value of the assets has diminished over time.
Negative equity usually occurs in the real estate market with mortgages. When the price of the house drops from a recession or some other reason, this causes negative equity. It often means that a home/land owner is actually paying more for the house/land than what the asset is actually worth. Also, if the borrower defaulted on the negative equity debt, then the lender would not recoup the amount given simply through repossession.
If the loan is non-recourse then it means the creditor will only be able to recoup some of its costs through the sell of the asset. This type of equity has also been known to occur if the value of the collateral asset stays fixed, and the value of the loan payments is less than the interest of the loan. Refer to this negative equity as a negative amortization.