Outlook all of a sudden has begun asking for a password, even though the "remember password" option is checked. What's worse, it rejects the password when I type it in - so no e-mail for me. Help, please!

Are you remembering to uncheck the "remember password" option before you type in the correct password?
You need to uncheck that option, clear the password, then type in the correct password. If it works, then you can recheck the "remember" option.
If not then you need to double check what the correct password is. You, or another user, may have changed it and forgot about doing it.

   Outlook will ask you for a password if the password that it has did not work.  I think dlwolff0 alluded to this in the last paragraph of his post.
   If the password that you are entering is correct, the problem could be with your email provider.  Sometimes email providers have temporary problems.  If this is the case, it should start working eventually.  If it doesn't, contact your email provider and ask them why the password is not working.