Some app categories benefit from Apple's push notifications more than others. Twitter, along with chat apps, stand to benefit the most. Here's your definitive list of all Twitter apps featuring push updates or alerts.
Twitbit is a full-featured Twitter client allowing you to receive push notifications for mentions and direct messages. Unlike iTwitter (below), Twitbit doesn't require both users to be using the same app.
Boxcar checks your Twitter account every two minutes for mentions and direct mentions. If you get one it will push you the message and let you see it in your preferred Twitter client. Still not instant, as Twitter really should be, but it looks to be the best solution at this time.
Like Boxcar, Tweet Push is not a full-fledged Twitter client. Instead, it pushes your mentions and DMs and pops open your preferred Twitter client (Twittelator, Tweetie, Twitterfon or Twitterrific). However, unlike Boxcar, this one is subscription based. It'll cost 99 cents every month for each Twitter account you want to use it on.
Also pushes you Twitter replies and direct messages like Boxcar and Tweet Push but gives you a little more of the functionality you would expect in a Twitter client.
iTwitter is a first full-fledged Twitter client with Push Notifications but comes with one major catch - you both have to be using iTwitter to receive the notification. You can receive mentions and direct messages.
A chat application that supports a ton of popular services including Skype, Facebook, Yahoo, AIM, MSN, ICQ, Jabber, Google Talk and MySpace. It also logs into and pushes your Twitter account and supports more services than BeejiveIM.
Another Twitter client that pushes your mentions and DMs. Unfortunately, it features a couple bugs in the 3.0 release that should probably get worked out before you consider buying it.
If you get Growl alerts for Twitter on your desktop Mac you can get those alerts pushed straight to your iPhone or iPod Touch. Not the most elegant solution, but it's available.