I've now got the ebay and messenger apps installed and working - and both of these seem to run in the background (I've had pop up alerts appear at the top of the home screen either when an item on ebay is ending soon, or a friend sends me an IM on messenger).
I noticed on the messenger app that if I use the 'back' button to exit the app, it asks me to confirm if I want to sign out - so I'm guessing that if I use the Windows key to go back to the home screen, the app is still running in the background.
This is called "toast" notifications. The application has closed (any third party app not actually open and visible is really closed), but ebay's servers can send Microsoft the ebay item notification, which they in turn send to your phone. The application is completely unaware that the notification has arrived, and is still closed. WP7 handles it all and if you click on the toast it should open the application.
The reason it asks you to confirm sign out when you click back is because WP7 has two ways to launch an app. When you return to it, e.g. by pressing the back key from another app, the app is allowed to pretend they were running in the background. It wasnt really, but they reload all the information and you cant tell the difference. If you click the Back key in the app, they are forced to close. If you click the app in the start menu, they are forced to launch as new.
So, there are some things we cant do at all. e.g. awarner's alarm app is impossible to do properly. If someone figures out how to build and install native applications it may be possible, but you still wouldnt see them in the marketplace. To be fair to Microsoft though, you were fooled with the ebay app and I think most reasons people think they need multitasking can done this way instead while using no extra resources in the background.
Just to confuse matters there are some third party applications that can run in the background. They have to be pretty big partners though with very good reasons, like Spotify.