It would be a lot easier if milk was just milk, flour was just flour and sugar was just sugar. But with variety comes the ability to make exactly what you want in the kitchen, with the right ingredients for the job.
With varying degrees of colour and flavour intensity, the type of brown sugar you should choose to use depends entirely on what you want to do with it. The darker the sugar, the more of a deep, rich, earthy molasses flavour it will have. Since molasses is intense and can easily overpower other ingredients in a recipe, some recipes call for one type of brown sugar over others.
Generally speaking, “golden” brown sugar has a light, delicate flavour without the heavy molasses notes of darker brown sugars. Regular brown sugar is dark and moist and is used for tasks where you want more of a molasses kick. Demerara sugar is darker still, with large crystals that give it a crunchy texture.
Unless otherwise noted in a recipe, you can use one type of brown sugar in place of another, as long as you remember that using a darker sugar will have a more pronounced effect on the overall flavour, and that the texture of baked goods may differ depending on which you choose. All can be used for things like sweetening coffee, where a rich and sticky demerara can elevate an ordinary cup of dark roast coffee into something quite extraordinary.
Don’t confuse brown sugar for “raw” or “plantation” sugar, which is typically not soft. Using this in place of brown sugar in baking will change your results.
Some natural foods stores carry unrefined brown sugars, including types like muscovado. While these can be delicious, recipes may not turn out exactly as expected, either due to a different texture or a lower level of sweetness in the same volume of sugar. If you plan to bake with these sugars, it may take some experimentation in the kitchen unless the recipe was specifically designed for their use.