ACTUALLY.. This is brand new mind you.. On the newest version of OS X (Mavericks) you CAN send push notifications from a webpage to the desktop. But according to the documentation, NOT iPhones:
Note: This document pertains to OS X only. Notifications for websites do not appear on iOS.
Currently Apple has plans to allow 2 kinds of push notifications:
OS X Website Push Notifications
and Local Notifications.
The obvious hurdle here is that this will not work on PCs, nor will it allow you to do android push notifications.
Furthermore, you actually can with versions as old as Snow Leapord, send push notifications from a website as long as said website is open and active. The new Mavericks OS will allow push notifications even if the site isnt opened, assuming you have already given permission to said site to send push notifications.
From the mouth of Apple:
In OS X v10.9 and later, you can dispatch OS X Website Push Notifications from your web server directly to OS X users by using the Apple Push Notification service (APNs). Not to be confused with local notifications, push notifications can reach your users regardless of whether your website or their web browser is open…
To integrate push notifications in your website, you first present an interface that allows the user to opt in to receive notifications. If the user consents, Safari contacts your website requesting its credentials in the form of a file called a push package. The push package also contains notification assets used throughout OS X and data used to communicate to a web service you configure. If the push package is valid, you receive a unique identifier for the user on the device known as a device token. The user receives the notification when you send the combination of this device token and your message, or payload, to APNs.
Upon receiving the notification, the user can click on it to open a webpage of your choosing in the user’s default browser.
Note: If you need a refresher on APNs, read the “Apple Push Notification Service” chapter in Local and Push Notification Programming Guide. Although the document is specific to iOS and OS X push notifications, paradigms of the push notification service still apply.