I've heard of zero G flights, but why exactly do we feel weightless when we are in them?
Suppose you are floating in space inside a rocket. You feel weightless because there are no forces on you.
Suppose you turn on the rocket engine. Your seat pushes you forward as the rocket accelerates, much like the seat of an airplane on takeoff or the seat of a car when you step on the gas. This causes the sensation of weight.
If you were standing on a scale in the rocket, the floor would push the scale forward (= up if you are standing), and the scale would push you. The scale has a spring, which would be squeezed between you and the floor. This is what causes the scale to register weight.
Standing on on Earth is very much the same thing. Gravity pulls you down. If you walk off the edge of a cliff, air doesn't hold you up. You fall, and you feel weightless.
When you stand on the floor, the floor pushes up on you hard enough to resist gravity. holds you up. The floor holding you up is what causes the sensation of weight.
If you stand on a scale, the scale is squeezed between you and the floor. The spring is compressed and the scale registers weight.
The rocket floor and earth floor situations sound kind of alike. In fact they are much more than kind of alike. Einstein realized that if you were in a closed room so that you could not see if you were changing speed, there is no experiment you can do that would tell you if you were feeling weight because of a rocket or gravity. From this insight, he developed General Relativity. For more about this, see my answer to this post.
Getting back to your question, zero G flights are like falling off a cliff. A normal plane flies straight and level. Gravity pulls you down and the seat holds you up.
In a zero G flight, the plane dives. As you fall faster and faster, the plane dives faster and faster. The seat doesn't hold you up any more than air does. Of course after 30 seconds or so, the plane pulls up and resumes normal straight and level flight so it doesn't crash.
A object thrown forcefully up into the air will follow a parabolic ("ballistic") trajectory, and a G-meter fastened to it will measure zero along that path (until it strikes the ground, of course) exactly as if the projectile had instead been dropped off a cliff.
An airplane, if following the right trajectory through the air, can achieve the same result- where it and everything inside it coasts up, over, and then down on that path, during which time a G-meter inside the plane will measure zero and its passengers will float freely inside it.
To achieve this, the pilot climbs the plane up to a high altitude and then dives to build up speed. The pilot then pulls up on the controls, putting the plane into a steep climb while watching the G-meter. When it reads zero, the pilot eases off the climb and works the controls to hold the G-meter at zero. The plane coasts up, noses over, and then heads down towards its original altitude.
In this way, the plane will furnish a minute or two of weightlessness to its inhabitants.