G-force is a measurement of force per unit mass that causes the perception of weight. Since it is a measurement of force per unit mass, which is actually acceleration, the term g-force is technically incorrect. It is a measure of acceleration and is equal to 9.8 $m/s^2$ at the surface of the earth.
You have the perception of weight simply standing on the ground due to the reaction of the ground to the force of gravity acting on you, even though you are not actually accelerating with respect to the ground. But you are said to be experiencing a g-force of 1 g upwards, even though the force of gravity acts downwards.
Conversely, if you were are in "free fall", as if you were in an elevator whose cables were suddenly cut, you would not have any perception of weight (i.e., you would experience weightlessness) even though you are accelerating with respect to the ground. You are said to be experiencing zero g.
If you stand in on a scale in an elevator at rest, the scale will show
your weight here on earth, under the acceleration of 1 g.
Correct. And you are said to be experiencing a g-force of 1 g upward.
But when the elevator is accelerating upward the scale will show a
greater weight, and I’m wondering why?
When the elevator accelerates upward the inertia of the person on the scale resists the upward motion so that the elevator floor and scale must push up on the person in order to accelerate the person along with the elevator. The scale has to push upward with extra force to accelerate the person's mass upward.
Applying Newton's second law:
where $m$ is the persons mass, $a$ the acceleration of person, and $F_N$ is the upward force exerted by the scale. Therefore
Therefore the upward force exerted by the scale is greater than the person's true weight of $mg$.
Hope this helps.