This morning I am unable to retrieve mail from my ISP and I have an "Account Error" advisory at the top of my Apple Mail page - in a little black triangle. The Preferences show that iCloud is online but I cannot get it to go offline and allow my ISP mail to be online.

Every time I designate my ISP account as the online account, iCloud takes over and designates itself as online.

Solution?

NOTE: My wife's ISP email, on the same machine but a different account, is just fine.

Update: I used Cocktail to rebuild the Launch Services Database, but that made no difference.

Update 2: And now, just as mysteriously as the problem arose, it has gone away. I went to fill my coffee mug and, when I returned, Mail had started working properly. Colour me confused, albeit relieved.

Never underestimate the power of coffee.

The triangle indicates Mail was unable to establish contact with your ISP's email server, almost certainly because that server was temporarily down and had nothing to do with Mail or anything else on your computer. Normally both your ISP's account and iCloud account would be online simultaneously so I am not sure what you mean by "...I designate my ISP account as the online account,…" I am not sure what you were attempting. Whatever, the fact is when Mail was unable to establish contact with the ISP's server it would quite correctly and appropriately revert to the account that it could contact, iCloud.

While you were refilling your coffee mug, the ISP server came back online and because Mail queries the servers periodically contact was re-established. The problem was solved by and at your ISP.