The red dot next to a user's name indicates that the user has marked themselves away all day in the Phabricator Calendar application. However, this is really non-obvious (several people have asked about this in the context of the Wikimedia instance).
The user interface should give some kind of explanation or hint. It would be enough (at least for me), if there would be a hover text (would be easy with the title tag) saying: "This user is away" or something else
It should also be mentioned that the grey dot next to a disabled user could have the same problem.
Original report: https://phabricator.wikimedia.org/T105616