Brown sugar is refined (white) sugar that has had some of it's molasses reintroduced. Light brown sugar contains about 3% molasses while dark brown sugar contains about 6%. When a cocktail recipe calls for brown sugar, I always use dark brown sugar. I like the extra bit of depth and body that the darker form of this sugar brings, especially in Tiki drinks using aged, dark, or Demerara rum.
Usually a recipe will just call for brown sugar. And normally this means light brown sugar. But I go for the gusto. I have not found a drink suffering as a result yet.
You can substitute light brown sugar for dark brown sugar in the drink recipes on this site if you wish. The difference in the flavor profile of the finished drink is small though notable.
When measuring brown sugar you want to pack it lightly into whatever container you are measuring it in be it a teaspoon or a measuring cup. You don't have to make a hockey puck out of it. Just tamp it down a little.
Brown sugar will take a while to dissolve. If you just add it to a shaker and shake then you'll probably only dissolve half of it. If possible you can mix it with one of the non-alcoholic ingredients in the recipe first - water, fruit juice, etc. Then proceed as the recipe directs. Often the recipe will specify doing this. Alcohol reduces the ability of sugar to dissolve so save the liquor for after this step.