Damage Reduction, also referred as DR, is a special defense value that applies to damage done to a ship's armor and sails. Structure does NOT have any DR. Before damage from a shot is applied, the current DR value for the side hit is subtracted from that value. In some circumstances it can reduce the incoming Damage to only one (1) damage per shot! (which is the lowest number, zero (0) isn't possible).
You have 10 Damage Reduction (DR). Enemy has 10 cannons, which deal 20 damage each.
Normally you would get 10 x 20 = 200 damage. However, 10 DR lowers this value to 10 x (20-DR) = 10 x (20 - 10) = 10 x 10 = 100.
10 DR reduced incoming damage by 100 in this example, that's 50% of the damage you would have received without DR!
[Damage = [Damage of Cannons - Target DR] x [Amount of Cannons]]
It behaves very differently from Damage Resistance (which lowers incoming damage by a percentage and not a definite number). When the armor value of a faction starts to drop, the DR value of that faction will drop in the same proportion.
% DR = (equal to) % armor (on the specific side or sails)
This information is provided by the update that PotBS Wikia is undergoing, by Captain Vuur.