When you upgrade your Shipyard to level 2, it provides a 15% weight decrease for all Forsaken weapons, armors, specials, and tactical modules. The upgrade does not decrease the weight of Draconian technology. For example, every Thud Cannon IV component you use will see a 15% weight decrease, while Strike Missile D51-L components will see no decrease. If you're not certain which components are Forsaken and which are Draconian, simply look up the component in your Intelligence Lab; it will show what type it is on the right.
Here is an example of a Forsaken weapon (Phalanx Anti-Missile II, weight 250 tons) with a level 2 shipyard that brings the weight down to 212 tons.
Here is an example of a Draconian weapon, which is almost the same weight (Hailstorm Anti-Mortar B, weight 240 tons), that does not receive any decrease in weight when added to a ship.