Your password issue is probably easy to fix.
Apple Mail, like most email clients, stores your passwords so that you do not have to keep entering them when you send or receive email.
You did not say which event triggers the error message, but we'll move on.
At a guess, I would presume that you changed your password recently. Even if you didn't, try this.
Open Apple Mail and then open Preferences.
Select the Accounts tab and choose the RoadRunner account from the list in the left column.
In the Right column, empty out the password field and enter the "correct" password.
Still in the Right column, click on the up/down arrows on the "Outgoing Mail Server" field and choose Edit SMTP Server list.
Select your Server and go to Advanced.
Empty out the password field and enter the correct password.
OK out of there and close Preferences.
Should be OK now.
Let us know
I have a problem where when I try to click on a new message it doesn't allow me to. I have to close the mail application and then reopen it. The other issue is when I use internet accounts it says the password is incorrect. I then go down to where I can type the password and it works. This occurs only on my iMap from roadrunner. I tried talking to Apple Care and they say it is a roadrunner problem. Roadrunner says it is an apple issue. I don't want to reinstall the operating system.