Asynchronous means that you can do one thing, while the system does something else. Neither has to wait for the other to finish what it is doing. To give you a couple of examples:
1. A friend of mine has an asynchronous car heater. When it is cold, he can remotely activate the heating system. So while he is getting dressed for work, his car is making itself ready for a comfortable ride.
2. Most of us have asynchronous washing machines. We can start them, and watch a movie while it finishes.
It is even worse with Web 2.0, nobody understands what that is - and reading O'Reilly's explanation does not help much. And, there is just one catch... There never was a Web 1.0.
The third problem is the black & white syndrome.
With Web 2.0 and AJAX there is tendency to focuses on the absolutes - either you are Web 2.0 company, or you not. There is no middle way.
This is a big problem, because that is like saying that you either drive a sports car or you are a looser. Well guess what, driving a sports car sure is fun, but not when you shop for groceries, pick up the kids from school, help a friend move, or when you want to bring your bike into the mountains - and about a zillion other situations where a sports car simply does not work.
"Only a Sith thinks in absolutes."
- Star Wars III: Revenge of the Sith