What areas of the MacBook Air should I keep clear? According to the User's Guide of the MacBook Air, it says:


  Set up your MacBook Air on a stable work surface that 
  allows for adequate air circulation under and around the computer. Do not operate 
  your MacBook Air on a pillow or other soft material, as the material can block the 
  airflow vents.


The only airflow vents I see are in the back right where the screen and keyboard meet. Thus, I'm wondering if covering the front 90% of the bottom of the MacBook Air with something soft such as a cloth (i.e. the part with no vents) will have any adverse effects on the cooling mechanism. The only thing I can think of is that the bottom of the laptop acts as passive cooling by having that little bit of air between the bottom of the laptop and the desk its sitting on. Having it on a cloth or something similar would no longer provide this air cushion and would affect the cooling mechanism of the laptop.

Is my theory about how the passive cooling works correct? Or is it fine to use the laptop on something such as a pillow as long as it doesn't cover the end that has the vents?
You are correct. While the major fan vent is in the back, it is important that air can circulate under the laptop. This is because the bottom side of the laptop can get hot, and the space between the laptop and the (hard) surface helps release that heat. A soft, squishy surface will just trap it and push it right back up into the computer.And a computer with only one fan vent needs all the help it can get.

That said, you should be OK using the laptop on something soft for most of the time. It's when the fan starts spinning that you really should move to something more solid.
To keep an eye on how hot your computer is,\ and how fast the fans are going, I use the free iStat Pro widget.

Hope this helps.