It varies depending on the source. King Arthur's unbleached AP flour is 120 grams
Arrowhead Mills organic AP flour is 120 grams for 1 cup.
Pillsbury flour is 124 grams in one cup.
Look at the nutritional information on the side of the package. It will usually list 1/4 C = xxx grams.
However, a lot of pastry chefs have their own values they've used to develop their own pastries. Joanna Chang uses 140 grams for AP flour and 150 grams for bread flour.
Rose Levy Beranbaum goes the full mile and lists the following for all-purpose flour
SIFTED = 3.5 OUNCES = 114 GRAMS
LIGHTLY SPOONED = 4.25 OUNCES = 121 GRAMS
DIP AND SWEEP = 5 OUNCES = 145 GRAMS
If I'm following a recipe with given weights (God Bless those authors), I follow their given weights.
If you go to King Arthur's recipe site, you can click a button and convert the recipe using:
cups
grams
ounces