The Internet is all about communication. Not so much organization, but definitely communication - email, instant messaging, forums, social networks, RSS feeds, news discussions, et cetera. All of that communication is hard to organize, and Mozilla aims to change that.
They are demonstrating a new type of browser plugin that would be integrated into the browser. Dubbed "Snowl", Mozilla is showing off the pre-alpha software and pitching it as a way to seamlessly integrate communication into the browser just as easily as you do things like bookmarks today. The same navigational tools you could use on a web page you could in theory also use in communication, such as list views or continuous views for RSS feeds, message priorities and others. The software is still in such an early stage that there is no telling what all Mozilla would like to encompass with it, but the idea sounds pretty neat.
I don't quite see how it would be possible to facilitate real-time discussions, like IM clients or IRC, comfortably into a browser, but I look forward to what Mozilla has in store.