I noticed that in the area near my name, there is a dot. Mine is green, when nearly everyone else has a blue dot! Me, Puppet, and GWIM have it. What is the point of those dots?

the green dot tells you if that person is online in the forums at the moment
if it isn't green, then they are not on the forums right now

But what does my red dot mean?

mouse over it
it says that you are set as "invisible"