Nearly every site I use daily has an activity feed or something that represents a constantly changing group of infomation presented in a list like format. I first had the idea for River when I realized every activity on the web has four common properties: actor, verb, object & context. "Susie posted a commented on your photo" Here 'Susie' is the actor, 'posted' is the verb, 'photo' is the object and 'your photo' is the context. Turns out there's even a spec for it.
The golden rule is any group of related activities can be a River: your Github feed, blog's rss, HackerNews entries about Ruby. A user can then subscribe to it and receive updates on that particular River's activities only. Or they don't have to be connected at all, such as a random River. Just like Pandora allowed users to create their own ratio statios, River allows users to create their own activity streams.