How do you change your Apple Mail password?

To change your iCloud mail password you have to change the password of the Apple ID used to sign into iCloud (which is probably the same as the other places you are signed in such as Music and TV) so everything will need to be signed out of and back in after you've changed it.

Please see Change your Apple ID password - Apple Support

Why do you want to do this? If you are getting frequent requests for the password please see the page linked to at the top of this page.

Web mail address and password can be used as an Apple ID to sign in Mac system preferences , apple mail is set when apple mail application is configured with iCloud and web accounts ( it could be gmail , yahoo ....) see the article Add or remove email accounts in Mail on Mac - Apple Support

So apple mail password is set with web mail account , it's not necessary an Apple ID .

Change the password on web account , it depends which mail services is being used .