How to remove the Mail icon indicator applet?

You can remove the message indicator by removing the indicator-messages package by clicking on that link and clicking uninstall. Or by executing the following command in a terminal:

sudo apt-get remove indicator-messages

Then log out and back in.

If Thunderbird Mail is installed and is the default mail application in system settings->Details: then start Thunderbird, go to menu->Tools->Add-ons->Extensions and disable Thunderbird extension "Messaging Menu and Unity Launcher integration". Quit Thunderbird. Logout and Login to Ubuntu.

This solved it on Ubuntu 14.04 and 16.04 LTS without the need to uninstall anything.

Remove the package indicator-messages from within the Ubuntu Software Center.

Then reboot, gone!

This instruction will not remove the sound icon on Ubuntu 12.04 (Precise)

The program can be uninstalled from your system by opening the program "terminal", and typing in

sudo apt-get remove indicator-messages -y

It will ask you for your password, and then it will remove the program from your system.