Apple's page has a nice summary of the feature:
OS X Mavericks introduces a powerful new way to keep users engaged with your website. You can now use the Apple Push Notifications Service to send notifications to your website users, right on their Mac desktop — even when Safari isn’t running. Safari Push Notifications work just like push notifications for apps. They display your website icon and notification text, which users can click to go right to your website.
Any chance Stack Exchange could add support for this, so new items in our SE inbox or new SE notifications could be pushed to Macs running Mavericks?