Loosely packed means you fill the measuring cup with sugar then use a spatula to remove the mound.
Lightly packed means you press the sugar lightly into the measuring cup and add more, press lightly down again, until the amount of sugar reaches the brim.
Firmly packed means you press the sugar more firmly, packing it in the measuring cup more tightly, adding more and repeating the process until you have a very compact amount of sugar.
To test whether your sugar has been loosely, lightly or firmly packed, fill a measuring cup with sugar then invert the measuring cup onto a plate or bowl. If the sugar does not retain the shape of the measuring cup, it is loosely packed. If the sugar partially retains the shape of the measuring cup (i.e., some come loose), it is lightly packed. If the sugar comes out as a solid mass (i.e., very little comes loose), it is firmly packed.
Does any of it matter? Yes, because the amount of sugar will vary and, ergo, the sweetness, color and texture of the baked product is affected. So, it does pay to use the correct measurement.