You can see the latest comments posted by the people you follow, and view replies to your own comments.
The "Now in your network" tab shows by default when you click on the "notification bubble" in any Disqus thread. This tab is comprised of the most recent comments posted by people in your network (users you follow). You can beef up your activity feed by following more users.
When another user replies to one of your comments, you will see a notification in your "Notifications" tab. To view the notification, click the Notifications tab. You will see a feed of all recent replies — new replies are indicated by a blue bar to the left of a user's avatar.