It seems like a no-brainer to me that this type of functionality would exist among the throngs of social networking applications. If the game where you flick the thing while the music is playing can tell me when I've got a new challenger from the Far East, why can't my favorite Twitter client let me know when I've got a new direct message? Sadly, most of the apps in the store (as far as I know) haven't implemented this functionality yet. I realize that it has some sporadic adoption, but certainly not the universal support that I personally expected.
Enter Boxcar [iTunes Link]. This little free app has existed in the store for months now and has received some very nice reviews, and for good reason. I've been a faithful user of this app for many moons and I find it pretty indispensable. Here's how it works:
When you install the app (which is free at the time of this writing), you can choose to enable a single service for notification -- Twitter, Facebook or an email account. It will poll the selected service at a short interval and send you a standard push notification message if there are any updates. In the case of Twitter, you can configure it to open one of several supported Twitter client apps. If you want to enable additional services, you can do so through the in-app purchase mechanism, and most options cost $.99US.
I'm a big fan of Boxcar and use it for Twitter and Facebook. It's one of those free applications that's actually worth more than what you're paying for it -- a quality that's becoming increasingly rare among free iPhone apps these days! Version 2.1 was just released a few days ago and includes several new features like support for Facebook Mail and better accessibility.
Pick up Boxcar now for free from the App Store.