The short answer is only 1 app runs at a time to prevent memory issues.
Apple will release a "push notification service" in September to allow 3rd party apps to receive updates in the background, for example instant messages. Currently, in apps like AIM instant messages can only be received if the app is running, but in September instant messages can be pushed even when the app is not running.
There are certain tasks that can be done at the same time with Apple's apps. Listening to iPod, while writing email/SMS or surfing the web. Phone calls, email, SMS are received while another app is running.
Some 3rd party apps are adding features to save/resume when quitting and returning to an app, for example Pandora streams music and (just recently updated) can now resume to the song playing before the app was quit.