Thunderbird Extensions, Minimize Thunderbird to the Tray
"Extensions" are mini-programs that extend Thunderbird's basic functionality. Thunderbird comes without extensions, (perhaps with the exception of Talkback, a crash reporting utility.)
( To minimize Thunderbird to the system tray on Windows or Linux, please see below )
You can obtain extensions in much the same way as themes, (see the Thunderbird Themes Tutorial.) To browse extensions available for your version of Thunderbird, go to Tools > Extensions, and the Extensions dialog will pop up. Click the Get More Extensions 
link, which points to a secure area of the Mozilla website from which extensions can be downloaded.
Like Themes, some Extensions do not support all versions of Thunderbird. It is not dangerous to install an obsolete extension, it will simply waist your time, (by refusing to work.)
Mozilla Thunderbird Extensions
Minimize Thunderbird To The System Tray
Minimizing Thunderbird to the system tray is possible for Windows and Linux.
Minimize Thunderbird to the Tray (Windows)
The Thunderbird extension Minimize To Tray lets you minimize Thunderbird (as well as the Mozilla Suite & Firefox) to Windows' system tray.
Before downloading this extension, you may want to consider the ThunderTray Utility ("TB-Tray"), which includes several nice features, in addition to minimizing Thunderbird to the tray. (Update: the ThunderTray website no longer exists, see the link below instead.)
The Minimize To Tray extension can be downloaded from the Mozilla website:
Minimize Thunderbird to the Tray (Linux)
To answer a common question: Yes, you can minimize Thunderbird to the tray on Linux, but not with the Minimize To Tray extension (at this point).
To minimize Thunderbird to the tray on Linux (Gnome, KDE, XFCE 4, Fluxbox and WindowMaker), download Alltray. Alltray is not a Thunderbird extension, but allows you to do the same - for all apps.