Users don't need to know how cloud computing works. We only need to know what it does for us. This is seems to be the approach to explaining it that the industry itself is taking, and that is all well and good to me. I am a writer by trade, and I chose that path because I am not a very technical person.
The simplest solution is the best one says Occam's Razor. If someone asked you what an iPod was it would be best to just show them yours. Cloud computing is similar. If you have ever used Google Apps, like Google Maps or Google Docs, you are using cloud computing. PayPal, Facebook, Salesforce, and Amazon Web Services are also examples of cloud computing. Cloud computing allows access to "software as a service" that exists on the Internet, rather than being unique to your computer. If you need something other than your computer or mobile computing device to access an application, it is not using cloud computing.
A big buzzword in the field of cloud computing is "scalability." This means that computing resources exist in the cloud that can meet your needs as they scale up or down and you pay for them accordingly. You won't find yourself waiting around for financing to buy additional servers when you need the space, and if things slow down, you won't be stuck with hardware you aren't using. So cloud computing is relevant to you if you need to know how to get to the dentist across town or if you need to process unlimited quantities of information quickly and efficiently.

References:
Cloud Computing Journal
Wikipedia
Salesforce
Google Apps Software as a Service
Horizon Report