What is the equation used to convert grams into newtons and newtons into grams?

1g of weight = 0.001kg of mass * 9.80665m/s^2 (gravitational constant at sea Level on earth)

so...

1g = 0.00980665N
1N = 101.972g

Grams can't be converted into Newtons. A Newton is a force. A gram is mass. Force isn't mass.

As jcaron2 mentioned, the force (weight) is dependent on the acceleration due to gravity but on the moon or somewhere else (where the acceleration due to gravity differs), the weight (force) is different.