A Design Run is a rough full run of the show on the stage for the benefit of the designers. This should happen after Initial Hang And Focus but well before Tech Week.
All designers and the Master Builder should be present. They should be looking at their area of emphasis and taking notes of any corrections/tweaks and anything that does not look/sound right.
Immediately following the run, there should be a Design Meeting (no actors present) to discuss what happened. This is an opportunity for the designers to adjust their aspects of the show and communicate with the Director with a run fresh in their mind.