Green with play = server is running
Red with stop = server is not running
Gray circle = unknown, being a remote server management studio hasn't tried to connect to it yet, thus doesn't know if it is running or not, I think this icon is supposed to be limited to the registered servers window.
There seems to be a slight bug though where a connection is made without its green icon and only shows the empty gray circle. I normally see this when connect to a server that has a gray circle in the registered servers list, it then appears in the object explorer with a gray circle and all other connections after that will also have a gray circle, ah well.
I don't think the managment studio has any way to tell between local/remote servers.