It wasn't long after push notifications were finally introduced in iOS 3 that frustrations began popping up right alongside them. The blue alert dialog that appears to let you know you've got a new text message or update you on your sports scores is handy and all, but it can induce plenty of annoyance when you're in the middle of doing something else. Plus, if you dismiss the notification just to get it out of the way, there's no way to get back to it later.
iOS 5 improves notifications in three key ways: the introduction of banner notifications, the addition of Notification Center, and improvements to lock-screen notifications.
Banner notifications solve the problem of having notifications interrupt everything you're doing. Instead of appearing as a dialog box, the top of the screen flips down—like one of those rotating billboards—to reveal a small icon from the notifying app and a message. Tapping on the banner takes you to the app in question; but if you don't tap the notification, it will linger for a few seconds and then disappear again.
But what if you don't quite tap fast enough? No worries—that's where Notification Center comes in. Just swipe down from the menu bar and you'll drag down a sheet containing all of your recent notifications, arranged by the originating app. As with the banner notifications, tapping anything in Notification Center will take you to that app (and usually to the relevant item).
You can clear all the notifications for any app by tapping the 'x' icon opposite its name, then tapping the Clear button that appears. However, there's no way to clear just a single notification without tapping on it—my experience with iOS's multitouch conventions tells me I should be able to swipe across a notification and have a Delete button appear, but sadly that doesn't appear to be the case; it's an all-or-nothing proposition.
Notification Center gives you two choices for organizing the order in which apps appear in it: manually or by time. If you choose by time, the app with the most recent notification will show up at the top, followed by the app with the second most recent, and so on. If you opt for manual organization, then the apps will appear in the same order that they do in Settings -> Notifications. You can reorder them there by tapping the Edit button and dragging them into your desired sequence.
In addition to notifications, Notification Center on the iPhone includes two special entries: a Weather widget and a Stocks widget. The Weather widget displays the current conditions and temperatures for the first location in the iPhone's Weather app (if you select the Weather app's new Local Weather option, that's what will show up in Notification Center). Tap on the widget and you'll be taken to the Weather app.