Firefox 17 brings a new Social API, which allows sites to do things like send notifications to users, even when they're not currently browsing the site:
Much like the OpenSearch standard, Mozilla’s new Social API lets social services integrate into Firefox in a compelling way and makes it easier to use the Web the way you want. The Social API includes features like a social sidebar, toolbar notification buttons and the ability to easily chat with your friends no matter where you are on the Web.
Currently, only Facebook has an example of this functioning. I'd like Stack Exchange to add support for this, so new inbox items and new notifications could show up in Firefox via this API.
Note: It looks like SE already supports some similar features in Webkit browsers: How is desktop notification implemented in chat.stackoverflow.com?