Q: why doesn't my mail icon open?

If you are referring to the Mail icon in the Dock, remove it from the Dock and then navigate to your Applications folder and drag the application to the Dock. Apple explains this in full in Mac Basics: The Dock holds your favorite apps, documents, and more - Apple Support

Quit Mail, then relaunch it by clicking its icon in the Dock while holding down the shift key.

Linc Davis wrote:

 

Quit Mail, then relaunch it by clicking its icon in the Dock while holding down the shift key.

Not trying to be argumentative but it works perfectly, please re-read the instructions and retry.