Hamachi uses colored shapes and indicators to portray peer/network status.
Stars vs. Dots
A glow/outline around the green indicator means that traffic is actively moving accords the connection.
Means that the peer is offline, and no connectivity is present. This would happen if the peer was not signed into Hamachi, or not online in the network. The indicator will also be gray if you are not online in the network.
Cyan indicators mean that Hamachi was unable to establish a peer to peer connection, and fell back to using a relay.
Means that both peer to peer and relay connection attempts failed. This is usually caused by a network firewall or software firewall, sometimes by being behind multiple NAT devices on each end of the connection or by using the client that does not support relaying; most notably - 0.9 series of Linux and OSX clients.