A liability is an obligation owed by a company.
On the balance sheet, a company lists many different types of obligations. Things like accounts payable (money owed to its suppliers), debt such as borrowings from a bank, and deferred revenue (money received for goods or services to be delivered in the future). Each of these the company must satisfy in the future, either by paying cash, delivering goods, or some other settlement.
Liabilities can be either short term (generally defined as owed within one year) or long term (generally defined as owed beyond one year). The former are "current" liabilities, while the latter are classified as long-term liabilities.
While debt is one type of liability, not all liabilities are classified as "debt." The two terms are not interchangeable.
Difference between liability and debt
Debt is money borrowed from someone that has to be paid back, plus interest. You probably have this in the form of a mortgage or car payment or credit card payment. Companies have them in the form of mortgages, bonds, and bank loans.
A liability is something owed to someone, but is not necessarily debt. Accounts payable, for instance. The company bought some equipment or supplies and has 30 days to pay. A personal example might be the meal you eat at a restaurant, while you eat it and before you pay for it. During that (short) time, it is a liability. You've gained a benefit, the meal, but haven't paid for it.
Accrued expenses are another example. This one's easy (okay, it was a stretch with that meal example). You use electricity supplied by a utility, but only pay for it once a month. Employee salaries are another example. They've worked and need to be paid, but it isn't pay day yet.
Unearned revenue is a third example, though not all companies have it. This is where somebody has paid for something, but the company hasn't delivered what it has promised, yet. Think of a gift card. You've given Borders (for example) $25 cash, but they haven't delivered any books to anyone. Prepaid subscriptions are another example.
Thus, when looking at a balance sheet and seeing if the company has debt, don't confuse these other liabilities as debt.
Various liabilities are used to help determine the liquidity of a company. For instance, the current ratio looks at current assets (such as cash and accounts receivable) and compares that to current liabilities. If the ratio is greater than one, then the company has enough resources on hand to cover its immediate obligations.
The basic equation of accounting
Assets = Liabilities + Equity
On the balance sheet, the sum of the total liabilities and the shareholder's equity will always equal the total amount of assets. This implies that the liabilities are part of what is used to fund or pay for the assets.