While logged in, with the app open, you will automatically receive notifications from the repositories that you watch:
Notifications will be shown for the usual things, like issues and pull requests being opened/closed/commented on, as well as git events, like commits being pushed, or branches and tags being created/deleted. Clicking on any notification will open up more details on GitHub.com.
Update: We’ve since removed notifications from GitHub for Mac because the feature was kind of half-baked, and we prefer not to use half-measures. We might reintroduce it in a future release after we’ve had some more time to think about how it should work. In the meantime, we recommend that you use the website notifications or email notifications.