One of the most welcomed features in iOS 5 is the addition of a notification center. It's one of the things I found myself jailbreaking older iPhones in order to have. The notification center finally makes push notifications worth utilizing.
There are three flavors of notifications: Banner-style messages strip across any app you're using, including the phone's lock screen. You can also make a downward swipe from the menu bar to reveal full list of notifications. Finally, alerts are the familiar, large blue screens that pop up and demand action before being dismissed.
The banner-style notification is the one that I used the most in my testing. It was set up to push new mail messages, texts and BeeJive messages to me. If you tap the notification in the banner, it'll switch you to the app that sent it. They go away after about five seconds or so. Sometimes, you'll see several notifications at once if you're reconnecting to the Internet after being away for a while.
The notification center itself contains all these notifications and more, including widgets for stocks and weather. There are no third-party widgets supported at this time, and I hope that Apple opens up the Notification Center's API in the future to foster development.
You can customize a good deal of your apps to appear in the notification center as well as the style of notification each app uses. You also can get rid of the stocks or weather widget if you don't care to see then. Go into Settings, then Notifications and turn them off. This is where you can customize notifications for other apps and the style of notification that is received. You can also set sounds and whether or not the notification appears on the phone's lock screen.
Notifications are among the best of the new features in iOS 5, and it's worth taking about 15 minutes to fiddle with the settings to get them to your liking.