'OS X Mail keeps asking for iCloud password' Solution

Enter password for account "iCloud"

This is soooo annoying to me. I've finally figured out something that works.

Open Mail

Go to Preferences (Mail > Preferences)

Go to Accounts

Select your iCloud account from the left.

Choose the Advanced tab

Accounts > iCloud > Advanced > Authentication

Change the Authentication setting from "Apple Token" to "Password".

Change from 'Apple Token' to 'Password'

Close Preferences and click "Save" when it asks if you would like to save your changes.

Voila!

Haven't had it ask for the password since and it was happening ALL THE TIME for me. Hope this helps someone else!