There are a couple of causes for the password prompt appearing:
1. Password is incorrect or has been changed.
You can login to the control panel and confirm the password for individual email accounts from the Mail Manager.
Also make sure the full email address is set as the username, including the @ symbol.
2. More than one computer is accessing the same email account at the same time.
POP Accounts can only have one login at a time to a particular mail account. So if two computers have the same account you will need to close down the email program on one of them when using the other.