"G-force" refers to the apparent inertial force that is "felt" by the pilot and the other contents in an aircraft as it accelerates. For example, if an aircraft is accelerating upwards at the start of a loop, the pilot "feels" an apparent force pulling him downwards into his seat more strongly than usual.
So G-force describes the mirror-image of the force exerted on the contents of an aircraft by the aircraft due to the acceleration of that aircraft.
For a force to be "felt" by the pilot, the aircraft must "push" against the pilot's body, creating stresses and strains within the pilot's body. The ultimate origin of these forces that the pilot "feels" is the aerodynamic forces created by the wings and other surfaces of the aircraft, and the thrust force created by the motor. These forces are transmitted as stresses and strains through the structure of the aircraft to the pilot's seat, seat belts, etc and then are transmitted as stresses and strains to the pilot's body, as well as to all the other contents of the aircraft.
In other words, the G-force applied to an object within in aircraft as an aircraft maneuvers could be said to be equal to the net vector sum of all the aerodynamic and thrust forces created by the aircraft, divided by the ratio of the mass (or weight) of the object in question to the total gross mass (or weight) of the aircraft. Visualized this way, the units of G-force would be Newtons in the metric system, and pounds-force in the English system.
Normally however we talk about G-force by referring to the acceleration that would result from that force, but rather than using the usual units of acceleration, we use the units "G's", where 1 G is the acceleration caused by earth's gravity on an unrestrained object in a vacuum near the surface of the earth. Expressed this way, the so-called "G-force" (which really would more properly be called "G-acceleration") is equal to the net vector sum of all the aerodynamic and thrust forces created by the aircraft, divided by the ratio of the mass (or weight) of the object in question to the total gross mass (or weight) of the aircraft, divided by the weight of the object in question.
Note the intimate relationship between the G-force or G-acceleration, and the actual aerodynamic and thrust forces generated by the aircraft. (This is why we use a G-meter to avoid generating too much lift and pulling the wings off an aircraft during aerobatic maneuvering!) If the pull of the earth's gravity vanished, but the actual aerodynamic and thrust forces generated by the aircraft somehow stayed exactly the same, the G-force or G-acceleration would also stay exactly the same. (The trajectory of the aircraft, of course, would change.) So in this sense, the G-force or G-acceleration actually has nothing to do with the pull of the earth's gravity.
Another way to understand G-force or G-acceleration is to realize that the G-force or G-acceleration is the mirror-image of the "felt" component of the actual net force acting on an object or the net acceleration experienced by an object. Gravity can't be felt, because the force of gravity exerts an equal pull (per unit mass) on every molecule of an aircraft and contents, and so works from "within" to accelerate the aircraft and all the contents as a single entity without creating any stresses and strains within the aircraft and its contents. So the acceleration component resulting from gravity is not "felt" by a pilot's nervous system, and does not deflect a G-meter, scale, or other similar instrument.
A few examples make these concepts more clear:
An imaginary aircraft in freefall in a vaccuum. Actual acceleration is 1 G downward, but "felt" acceleration is zero. Pilot and passengers float weightless. Note that there are no aerodynamic or thrust forces, so the vector sum of the aerodynamic and thrust forces is zero. G-acceleration (normally called "G-loading") is said to be 0 G.
"Vomit comet" aircraft in 0-G trajectory. The wing is unloaded to the zero-lift angle-of-attack, and engine thrust is modulated so that it is always exactly equal to drag. So the vector sum of the aerodynamic and thrust forces is zero throughout the maneuver. Pilot and passengers float weightless. Note that while the "felt" acceleration is zero, and the G-loading is said to be 0G, the actual acceleration is 1-G downward. The aircraft is accelerating straight toward the earth down at 9.8 meters per second per second throughout the maneuver.
Spacecraft beyond atmosphere. There is no aerodynamic force. Whenever the motor is not firing, the vector sum of aerodynamic and thrust force is zero, so the occupants experience zero-G or "weightlessness", regardless of the actual trajectory of the aircraft, i.e. regardless of the actual strength of the gravitational field around the spacecraft.
Aircraft in straight-and-level flight, at a constant airspeed. Thrust and drag are equal and opposite. The net acceleration is zero. Gravity is creating a 1-G downward acceleration component which is "unfelt", while the lift from the wings is creating a 1-G upward acceleration component which is "felt". The G-acceleration or G-loading is the mirror-image of the "felt" component of the net acceleration, so the G-loading is said to be 1-G downwards.
Aircraft at rest on the ground. The net acceleration is zero. There is no aerodynamic or thrust force, but the earth is pushing up against the wheels of the plane with a force that would create a 1-G upwards acceleration if unopposed. Just like aerodynamic or thrust forces, this force creates stresses and strains that are transmitted through the aircraft structure to the pilot's body. The "felt" acceleration is 1-G upwards, so the G-loading-- the mirror-image of this "felt" acceleration-- is said to be 1 G downwards.
To experience a reduced G-loading in flight, all you have to do is firmly push the control stick or yoke forward to reduce the angle-of-attack of the wing. This reduces the lift force, and thus the G-loading. As the flight path curves earthward, the airspeed will rise, so you'll have to keep pushing the stick or yoke further forward to further reduce the angle-attack if you want to keep the G-loading low. Sooner or later the process will become unsustainable as the aircraft approaches Vne-- sooner if you start from level flight, and later if you start from a steep climb-- but until that point, you'll get to experience a reduced G-loading. The seat will push up against your body with less force than in normal straight-and-level flight.