The green "Check" listing tells you what setting you should check or have ticked.
The red "Uncheck" tells you what should be uncheck or have unticked.
General:
1. Go to Admin -> Configuration -> Features and Options -> General.
Check:
Enable who's online list - allows you to see who is doing what and where.
Track daily statistics - tells you stats about your users.
Track daily page views (must have stats enabled) - lets you know how many users were viewing specific pages. (Optional)
Uncheck:
Allow users to edit their displayed name - you can keep a better track of who is who if they keep the same user name. Let them request it and keep a log of it somewhere so you and your staff knows. I suggest creating a "Changelog - Users" either in the admin or staff boards.
Allow non-administrators to hide their online status - don't let others hide their status. (Optional)
Layout:
1. Go to Admin -> Configuration -> Features and Options -> Layout.
Check:
Show online/offline in posts and PMs - You can still see if someone is online by viewing any of their posts, if they have any. This will work better than viewing Who's Online.