A negative liability typically appears on the balance sheet when a company pays out more than the amount required by a liability. For example, if you were to accidentally pay a supplier's invoice twice, the first payment would reduce the original liability recorded in accounts payable to zero, while the second payment would have no offsetting liability, resulting in a negative liability on the balance sheet.
Negative liabilities are usually for small amounts that are aggregated into other liabilities. They frequently appear on the accounts payable register as credits, which the company's accounts payable staff can use to offset future payments to suppliers. Technically, a negative liability is a company asset, and so should be classified as a prepaid expense.
Most negative liabilities are created in error, so their presence indicates problems with the underlying accounting system.