There are two status colors that can be seen on a players avatar.
Green: Member is currently online on the main site. On their avatar you will see a green square in the lower right, and on their profile page you will see a green button that says 'Online.' Click the button to challenge them!
Red: Member is currently playing in a Live game. On their avatar you will see a red square in the lower right, and on their profile page you will see a red button that says 'Playing.' Click the button to watch them!
If there is no icon on a player's avatar, then they are not currently online.