It depends what you have in the cup: A cup is a measure of volume, but a gram is a measure of mass.
Depending on the density of the material you are going to have in the cup, the weight in grams of the material will vary. For example, if you fill the cup with lead, the cup will weigh much more than if you fill with feathers.
This question would be similar to asking how much a 6-foot person weighs. It depends on the person, of course. A cup of water will weigh more than a cup of flour.
For cooking, you can find tables that convert cups to grams and vice versa for a variety of food. Below is a link with a table for standard cooking ingredients. Make sure you look carefully and scroll down to find the correct conversion table specific ingredient you are using! The conversion is different for each ingredient!
Here are a few common foods and their conversion from cups to grams (notice how much the weight varies!):
Granulated sugar: 1 cup = 200 grams
Brown sugar: 1 cup, packed = 220 grams
Sifted white flour: 1 cup = 125 grams
White rice, uncooked: 1 cup = 185 grams
White rice, cooked: 1 cup = 175 grams
Butter: 1 cup = 227 grams
Almonds, slivered: 1 cup = 108 grams
Oil: 1 cup = 224 grams
Maple syrup: 1 cup = 322 grams
Milk, non-fat: 1 cup = 245 grams
Milk, sweetened condensed: 306 grams
Broccoli, flowerets: 1 cup = 71 grams
Raisins: 1 cup, packed = 165 grams
Milk, dry: 1 cup = 68 grams
Yogurt: 1 cup = 245 grams
Water: 1 cup = 236 grams
Confectioners sugar: 1 C = 110 g
Cocoa: 1 C = 125 g