A *cup* is a unit of volume, not a measure of mass/weight. So the grams will vary depending what you are measuring.
A cup is 8 fluid ounces - not dry ounces. This means that if you fill it with water, you will have put in exactly 8 ounces of water. If you fill the same volume with other items, or dry ingredients (e.g. rice, raisins, marshmallows) they will not weigh the same, despite taking up the same space.
To be precise, an American cup is 237 ml... but I have a cup measure in my kitchen that they labelled 250ml. Maybe that's because I bought it here in Britain? But in the grand scheme of cooking, that extra 13ml (roughly a tablespoon amount) is a margin of error that shouldn't affect how your recipe turns out.
Here's a way to solve your weighing problem (I did this before buying a set of American cup measures):
Find a large tea cup or smallish beaker or tupperware container. Put it on the scale and zero it. Then fill it with water until it weighs 8 ounces. Notice how full it is -- the exact place where the water stops is the top of your "cup". If you can find a cup / container in which 8 ounces of water goes exactly to the brim - that's perfect for future use!
Tip out the water, and now you have a "cup" to use. No matter what dry ingredient you use, fill to that same line that the water reached. If you use butter, pack it in with a spoon so there aren't big air pockets. If you use brown sugar, pack it down with a spoon until it is level.
Another note -- flours are different here! If you are using an American baking recipe, it may take some experimenting to duplicate the look and texture of the same food made in America. If find that if a cake recipe calls for "all-purpose flour" then I double sift British "self-raising" flour before using it in the recipe to get a light and fluffy result. Otherwise, "plain flour" is all-purpose.
And a side note: a "stick" of butter is how the butter is packaged/sold in the States. It is a 1/2 cup. Or 4 ounces. Or about 115 grams. A stick is slightly longer and thinner than the standard butter packages sold in England. In fact, you buy boxes of "sticks" of butter. Here is a stick pic! lol http://whyfiles.org/shorties/075butter_pollut/images/small_stick.gif