How to disable thunderbird notification

Edit -> Preferences -> General -> "When new messages arrive:"

Unchecking the "Show an alert"-Box should do the trick.

This has changed in TB 3 and is now a bit trickier to turn off. You now need to set mail.biff.show_alert to false in the Config Editor as described at http://www.upfrontsystems.co.za/Members/izak/sysadman/that-annoying-thunderbird-notification-window. For Windows folks do "Tools/Options" rather than "Edit/Preferences" for the first step.

To turn off the notification sound, change mail.biff.play_sound to false. You can also get rid of the system tray icon by changing mail.biff.show_tray_icon to false.

Go to Tools->Options, then on the general tab, there should be a section at the bottom titled "When a new message arrives". Uncheck the box next to "Show an Alert".

See if that does it for you.