This can happen if you have lost your connection in the midst of communicating, so try toggling your Wi-Fi or other network connections off and then back on, to see if this helps the situation.

Sometimes odd problems with mail account servers can prevent authentication for a short while, so if this happens, consider waiting about 10-15 minutes before trying again.

Therefore, try again and ensure you have put in the right password.

Select any relevant ones (especially if there are duplicates), and delete them from your keychain.

Following this, go back to Mail and supply your password when prompted, and choose the option to save the password.

This may be especially pertinent if the password prompt is only happening when you try to send mail, or when you try to receive mail.

You might need to search on the Web or otherwise with your service provider to see what these settings are.

Lastly, try toggling the box to automatically detect and maintain account settings, to see if this setting may be causing the problem.

To do this, open Keychain Access and select your keychain, followed by choosing Keychain First Aid in the Keychain Access menu (or press Option-Command-A).

In the window that pops up, click Start to verify, and if you see any problems pop up (especially if they are related to your mail account passwords), then click the Repair radio button and again click Start.

Following this, you might need to re-supply your password again to save it.

And you should log out or restart once you remove the passwords from the keychain.

Technically, this shouldnt be necessary.

Losing the password irregularly, prompting PW entry again and again.

Mac Mail Asking For Gmail Password