There have been several articles posted about using  asynchronous pages in ASP.NET but none of them go into detail as to when you should use them. I finally found a great post by Thomas Marquardt that explains the process in depth.

He addresses a key misconception also:

So, in your ASP.NET application, when should you perform work asynchronously instead of synchronously? Well, only 1 thread per CPU can execute at a time.  Did you catch that?  A lot of people seem to miss this point...only one thread executes at a time on a CPU. When you have more than this, you pay an expensive penalty--a context switch. However, if a thread is blocked waiting on work...then it makes sense to switch to another thread, one that can execute now.  It also makes sense to switch threads if you want work to be done in parallel as opposed to in series, but up until a certain point it actually makes much more sense to execute work in series, again, because of the expensive context switch.

Pop quiz: If you have a thread that is doing a lot of computational work and using the CPU heavily, and this takes a while, should you switch to another thread? No! The current thread is efficiently using the CPU, so switching will only incur the cost of a context switch.

Ok, well, what if you have a thread that makes an HTTP or SOAP request to another server and takes a long time, should you switch threads? Yes! You can perform the HTTP or SOAP request asynchronously, so that once the "send" has occurred, you can unwind the current thread and not use any threads until there is an I/O completion for the "receive". Between the "send" and the "receive", the remote server is busy, so locally you don't need to be blocking on a thread, but instead make use of the asynchronous APIs provided in .NET Framework so that you can unwind and be notified upon completion. Again, it only makes sense to switch threads if the benefit from doing so out weights the cost of the switch.

Read more about it in these posts:

Performing Asynchronous Work, or Tasks, in ASP.NET Applications
http://blogs.msdn.com/tmarq/archive/2010/04/14/performing-asynchronous-work-or-tasks-in-asp-net-applications.aspx

ASP.NET Thread Usage on IIS 7.0 and 6.0
http://blogs.msdn.com/tmarq/archive/2007/07/21/asp-net-thread-usage-on-iis-7-0-and-6-0.aspx

PS: I generally do not write posts that simply link to other posts but think it is warranted in this case.