Underwater stock options have an exercise price which is greater than the market price of the underlying stock. For example, you may have options with an exercise price of $10 a share while the stock is trading at $8 a share.
For obvious reasons, you do not want to exercise underwater stock options, as you would being paying more for the shares than their current market price, and the exercise itself would not generate any tax loss that you could apply against other income. (Only in extremely rare situations might you purchase stock at a price that is greater than its fair market value: for example, if your company were privately held and thus you could not buy stock in the public stock market, and if you believed the stock purchase/exercise price would turn out to be much lower than the eventual price of any company acquisition or IPO—a big risk.)
See a related FAQ on the various approaches companies take to the problem of underwater stock options.