Brown sugar is moist and sticky, so its crystals leave air gaps when you spoon it in. Pressing it down removes those gaps. USDA data shows a packed cup weighs 220 grams and a loose cup only 145 grams, a difference of about 50%. Unless a recipe says otherwise, US recipes mean packed brown sugar.
The only difference is how much molasses is in the sugar. America's Test Kitchen reports that light brown sugar contains about 3.5% molasses and dark brown sugar about 6.5%. The weight per cup stays the same, and King Arthur Baking lists both at 213 grams per packed cup.
Brown sugar is a little heavier per cup than white sugar because it is packed. Coarse raw sugars sit on either side of it.
For white and powdered sugar in every cup size, see our sugar grams to cups chart.