If you are a Google Chrome user, you must have seen how notifications work for Gmail. Google created the Desktop Notification system based on HTML5, and proposed its inclusion in the W3C web standard. The draft for the notification can be seen on this page at W3C.
Notifications is designed to allow brief messages to be sent to a user however they are connected. Somewhere between email and IM lies Notifications.
Push notifications are a way for websites to send small messages to users when the user is not on the site. iOS and Android devices already support their own push notification services.
With this feature, Mozilla wants to bring the web-application closer to the desktop application. A similar experience on a desktop browser is a good start. Native mobile clients on Android have used push-notifications successfully, for a long time. However, the Push Notification system will also help developers send notifications to their mobile users, without investing in a separate mobile client.