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.

The above answer is no longer correct. As of Thunderbird 24.60, and probably earlier, the control is found here:

Tools -> Options -> General tab -> "When new messages arrive:"

Uncheck "Show an alert" box or use the Customize... button to alter the behavior of the notification if you wish.

Apart from new mail notifications (which are handled inside Thuderbird preferences), it seems that recent versions delegate the general notification handling to the operating system. In Ubuntu it can be found in Settings -> Notifications, where you can disable all Thunderbird notifications.