Provable fairness refers to the technology used by bitcoin casino and gambling sites to verify the outcomes of their games. This allows players to confirm that the outcomes were true and fair without any adjustments or manipulations by the gambling sites between when the bet was placed and when the outcome was revealed.
The basis of the system is the cryptographic hash function and it's difficulty to crack. Before the bet begins the gambling server generates a seed and send this as a hashed function to the player. Client side the player then generates a random seed and this is added to the hash function. After the bet is complete the player decrypts the hash function to determine that it is the same transaction that began. If the function matches then the bet has been uninterrupted. Any difference in the hash indicates that something has gone wrong with the outcome, most likely as a result of manipulation.