Damage Reduction (in short: DR) is a special defense value on the structure of the ship (Port/Starboard/Bow/Stern). Hull and Sails never have any DR. Before damage from a shot is applied the current DR value for the part hit is subtracted from that value.
It behaves very differently from % resist defenses because the % it lowers the damage by is dependent not only on the DR value but also the damage done by the hit. In some circumstances it can completely remove all damage done. When the structure of an area starts to drop the DR value of that area will drop in the same proportion. For example if the Port side of a ship had 1000 hp and DR 4 and takes a total of 500 damage the DR value of that area will now be only 2).