Tilde next to email account name...can't get or send email.

Tilde and No mail---Help?

One of my email accounts has quit getting email and it has a little tilde in a circle next to the account name (what does it mean?). I know that there is email there because I can go online to roadrunner and get the mail that way. I went under "window: connection doctor" and it has a green dot by it indicating ok. What could be wrong and how do I fix it??

One more data point that may or may not be relevant is that a few days ago the account suddenly started asking me for a password to get email. It goes to the box where my answer should be provided, but the box remains blank.