A two-finger swipe at the right edge of the trackpad brings up the Notification Center. It's designed to keep track of notifications and alerts from apps like Mail and Calendar through to third-party apps like Twitter. Facebook integration is coming later this year, and will be provided in a free update.
The Twitter sign-in process is the same in Mountain Lion as it is in iOS 5. Once authorised, you can send tweets from the Notification Center, as well as from the other core apps, using the Sharing button.