Your body feels acceleration in a funny way. When a coaster car is speeding up, the actual force acting on you is the seat pushing your body forward. But, because of your body's inertia, you feel a force in front of you, pushing you into the seat. You always feel the push of acceleration coming from the opposite direction of the actual force accelerating you.
This force (for simplicity's sake, we'll call it the acceleration force) feels exactly the same as the force of gravity that pulls you toward Earth. In fact, acceleration forces are measured in g-forces, where 1 g is equal to the force of acceleration due to gravity near Earth's surface (9.8 m/s2, or 32 ft/s2).
A roller coaster takes advantage of this similarity. It constantly changes its acceleration and its position to the ground, making the forces of gravity and acceleration interact in many interesting ways. When you plummet down a steep hill, gravity pulls you down while the acceleration force seems to pull you up. At a certain rate of acceleration, these opposite forces balance each other out, making you feel a sensation of weightlessness — the same sensation a skydiver feels in free fall.
If the coaster accelerates down fast enough, the upward acceleration force exceeds the downward force of gravity, making you feel like you're being pulled upward. If you're accelerating up a steep hill, the acceleration force and gravity are pulling in roughly the same direction, making you feel much heavier than normal. If you were to sit on a scale during a roller coaster ride, you would see your "weight" change from point to point on the track.
At the top of a hill on a conventional coaster, inertia may carry you up, while the coaster car has already started to follow the track down. Let go of the safety bar, and you'll actually lift up out of your seat for an instant. Coaster enthusiasts refer to this moment of free fall as "air time." More on that next.