You can uninstall or removes an installed awn-applet-quit-applet package itself from Ubuntu 12.04 LTS (Precise Pangolin) through the terminal,
$ sudo apt-get remove awn-applet-quit-applet
If you would like to remove awn-applet-quit-applet and it's dependent packages which are no longer needed from Ubuntu,
$ sudo apt-get remove --auto-remove awn-applet-quit-applet
If you use with purge options to awn-applet-quit-applet package all the configuration and dependent packages will be removed.
$ sudo apt-get purge awn-applet-quit-applet
If you use purge options along with auto remove, will be removed everything regarding the package, It's really useful when you want to reinstall again.
$ sudo apt-get purge --auto-remove awn-applet-quit-applet