Confused Guy asked: What are "air rights"? How could you sell the "air" over your building?
Thank you for the question, Confused Guy. First off, in most cases air rights come with land. You buy land and you own all the air above it -- just as one generally gets the rights to water and minerals below it.
Selling of air rights is a little tricky to explain -- and seems to apply only to New York (for now).
Basically, in New York, there are limits to how high you can build a building. However, if you want to build higher than the limits, you would need additional air space. You can acquire the right to that additional air space by purchasing it from someone. In other words, you would buy the air space above one property and put it atop your property allowing you to build higher than the limit allows.
Here's a simplified example: Let's say the building limit is 35 floors; but you want to build 70 floors. You own a space of land next to a flat parking lot. You own 35 floors worth of air above your land; the parking lot owner does too. You purchase the "air rights" over the parking lot and now have the right to build in 70 floors worth of air.
Some cases I encountered in researching this got much more complicated than the above example -- including one building that is going to stretch sideways into the air above neighboring buildings.