The black spot means they can’t chat because they were insulting or abusive in some way

The two red marks mean they are banned, meaning they can’t log in to the games because of abuse, cheating, or stealing accounts

Just hover your mouse pointer to the symbols and you will see

Hi Galaxy, 
The red mark signifies that this player is using a program. These players also have a computer icon on their profiles. if you click on it, it shows: "program usage detected". 
Some players who want to use a computer to decide their moves are honest and switch their accounts to a "computer" account. They also have an icon (which looks slightly different) on their profiles and the red exclamation mark. 
However, in most cases FlyOrDie's system detects the use of a program. This will be double checked by FlyOrDie administrators.
The use of a program in board games (only in board games) is allowed on FlyOrDie, but not in tournaments. 

The black dot next to a players nickname is a so-called demerit point.
Players who behave in an inappropriate way - who violate the **site rules -  receive a demerit point.  
A demerit point mutes the the player to all present in any of FlyOrDie's game chat areas.
The duration time of a demerit point varies. The player will be notified about the duration of the demerit at the login screen.

** site rules:   

https://www.flyordie.com/games/help/moderation.html

Have a nice day and good games!