Activity streams are lists of recent actions in the community. They include both your own actions and those of others.
The 'Recent activity' stream on your homepage provides an overview of what's been happening in all your groups, subscriptions and bookmarks.
If you scroll down the page, more activity will load and you can interact with this by liking comments, replying to them, or clicking on the question or blog to read more.
You will also find activity streams in any groups you've joined. However, these streams only show group-specific information.