Currently, Boxcar is a most excellent app for that straightforwardly pushes notifications from services like Twitter and Facebook to the iPhone. Boxcar 4.0 is going to make your iPhone feel like it has a million, constantly connected tentacles.
Boxcar 4.0 not only adds support for a ton more services—Google Voice, Buzz and others—the app's been completely redesigned so you can actually manage all of those notifications, neatly broken down by service with an overhauled dashboard. And, you can do stuff about those notifications inside the app now, like reply to a tweet.
The net effect of Boxcar is that the iPhone feels less siloed and more connected to the outside world, which is one of Android's biggest strengths. (The iPhone's push notifications system could still use a less obtrusive Android, Windows Phone or Pre-style redesign, though.)