A stream shows you real-time updates of activity as people interact with content, people, and places in the community.
You'll see an update in the stream when someone creates a document, replies to a blog post, posts a status update, and so on.
You can see all public activity (that you have permission to see) in the Most Recent stream. You can also create your own custom streams that show only the people, places, and content items you select. (Each person and Place in the community also has an activity stream: you can see yours under your profile, and Places have a place-specific stream that shows up on Place pages.)
Note that the News stream shows only new content. It does not show a stream update when someone modifies, comments, or replies to something. Content you've read will stay in your News stream for three days before going away. Content you haven't yet read will stay for seven days.
When you see the End of New Updates line in a stream, you'll know you've seen all the latest activity. A little further down, you'll see an End of Previous Updates line that shows you the updates from last time you checked.