I'm still not sure what app you're refferring to so I'm gonna guess its text messaging. I use Handcent instead of the stock msg app. It has a pop-up notification system that even allows a quick reply right from the popup. You'll find it in the android market Play Store.
There are a few apps that will work with your stock msg app to add popups as well. As for emails and everything else notifications are generally handled by the status bar, a pull down notification screen. If I'm not coming close to giving you an answer just be a little more detailed about how you'd like your phone to behave. If its been thought of, there's probably a way to do it with an Android phone. Like I said earlier, customization is one of the strengths of Android.