Unable to verify account name or password My Mac Pro asks for an e-mail and password.

I was unable to remember them; I got a new password, but I'm still receiving an error message:


  Unable to verify account name or password.
I suspect your old password is still in the Keychain; go to Applications > Utilities > Keychain Assistant. Search for your account (usually your email) or the server used. Then doubleclick on the entry and check the checkbox in front of Show Password. Verify that it's the new password which is stored there.
It's safe to delete multiple entries of the same account, you should just have one entry.

The background is that Apple Mail.app will first try and use the password(s) stored in the login Keychain and when those fail, it will prompt the user to enter the new password.

See this article on how to manage your Keychain.