"Air time" has a strange effect on your body because your body is not completely solid -- it is composed of many loosely connected parts. When your body is accelerated, each part of your body is accelerated individually. The seat pushes on your back, the muscles in your back push on some organs and those organs push on other organs. This is why you feel the ride with your whole body. Everything inside is being pushed around.
Normally, all the parts of your body are pushing on each other because of the constant force of gravity. But in the "free-fall" state of plummeting down a hill, there is hardly any net force acting on you. In this case, the various pieces of your body are not pushing on each other as much. They are all, essentially, weightless, each falling individually inside your body. This is what gives you that unique sinking feeling in your stomach -- your stomach is suddenly very light because there is less force pushing on it. The same thing happens when you drive down a dip in the road in your car or descend in an elevator moving at high speed.
On a roller coaster, this full-body sensation is complemented by all sorts of visual cues -- the upside-down turns, dizzying heights and passing structures. Visual cues are an important part of the ride because they tell you that you are going fast. Your body can't feel velocity at all; it can only feel change in velocity (acceleration).
The only reasons you know that you are moving quickly on a coaster is that the support structure is whipping past you at top speed, and the air is rushing in your face. Roller-coaster designers make sure to create plenty of tight fits and near misses to make you feel like you're rocketing through the structure at out-of-control speeds.
One of the most exciting elements in modern coasters is the loop-the-loop. These structures turn the whole world upside down for a few seconds. Let's take a closer look.