You'll find activity streams throughout NationBuilder. The biggest and most comprehensive of which is on your dashboard in the control panel. This stream gives you an overall, up to the minute view of everything happening in your nation: who is creating pages, who is tweeting about broadcasters, who is signing up on the website, who just left a voice mail, etc. More specific streams based around people and pages appear in other parts of your control panel and public website.
All of the activities that appear in streams can be edited when you go to Settings > Defaults > Dashboards in the control panel. Each column is a different dashboard in NationBuilder and you can control what streams a particular activity appears on by going to the activity and clicking on the edit icon. There, you'll see check boxes on the right side of the screen where you can choose what dashboards the activity appears on.
When you create new pages for your public website, you have the option to check Show stream in Settings for that particular page. If you add a stream to a page that stream is a public dashboard.
When Show stream is checked, as long as the activity is public, everyone visiting the site will see it. People with control panel access will always be able to see the activity, both public and private, in the control panel regardless of whether Show stream is checked.