I did the following and it started working again:
1) Right-clicked (control-clicked) on my .Mac/iCloud email account in Mail and chose "Export Mailbox..." to save it just in case.
2) Selected Mail...Preferences... in the menu and selected the iCloud account.
3) Clicked on the "-" (minus) button below the account list to remove the account and confirmed that I wanted to.
4) Clicked on the "+" (plus) button below the account list and entered my iCloud password.
5) Unchecked the boxes about contacts and calendars (those seemed to be working).
6) When I proceeded from there, the account came right back and all my emails downloaded back from the server.
All is good now.