g-force is just a term to describe the force required to generate 1 g of acceleration on an object. 1 g of acceleration is defined to be 32.174 feet / sec^2, or 9.80665 meters / sec^2. Note that this takes into account that the Earth is rotating, isn't perfectly spherical, and assumes the object is at sea level (average tide height) and at 45.5 degrees latitude.
The direction of the force doesn't matter, it's just a way of stating an accelerating force compared to the defined acceleration of gravity. In a Formula 1 race car, the cars generate downforce equal to their weight around 115mph, this could be considered "1 g" of downforce. At around 160mph, the cars can pull "4 g" turns. There's also about "1 g" of drag force at 160mph, so just lifting the throttle produces "1 g" of braking force, and with full application of the brakes, "5 g's" of iniital braking deceleration (while still close to 160mph).
Take away the Earth's rotation and assume it to be spherical, and g force increases to about 32.224 feet / sec^2 or 9.822 meters / sec^2. This definition of g-force is the base value used for space craft, (reduced by R/(A+R)^2, where R is the average radius of the earth, and A is altitude) since they move indepedently of Earth's rotation, and are not significantly affected by the non-spherical shape.