A repurchase right is a company's contractual right to buy back from an employee any stock resulting from the exercise of a stock option. The buyback can be at fair market value, book value, or the original purchase price. Typically, the repurchase will not be undertaken at book value or the original purchase price unless the repurchase is intended to act as a forfeiture provision.
One example of a repurchase right occurs if you have options that immediately vest and are exercisable but the company can buy back the shares (i.e., the repurchase right), often in a timeframe similar to a vesting schedule. Some pre-IPO companies grant these options subject to a repurchase right so that employees can exercise the shares when they have a low value and start the capital gains holding period.
See also a related FAQ about repurchase rights, and another FAQ about buyback provisions.