There is an integrated NIC that came with my Dimension 8300, and I think on other Dell models.  If it is enabled and no cable is connected, you get that icon. As I don't use the NIC, to stop that icon from appearing, I just disabled Intel Pro/100 VE Network connection in device manager. I disabled rather than uninstalled, so that it would be very easy to enable it again, should I ever want to use the ethernet connection. There are no side effects (other than the red cross in device manager, of course).

These will always pop up when the connection changes, but if you choose Always Hide icon in Taskbar customization, it might help. Also go into Network properties of each connection and uncheck "Show icon on taskbar when connected" for those you don't want to see all the time.

You should also be able to Disable the "Local area connection" using "Network Connections" , just Right click on "Local area connection" you want to disable & select "Disable".