Thanks for the help, although I should have posed the question better. Emails delete on the MacBook as they are supposed to, although some time ago, it all flowed back in multiple copies, a dozen or so at a time, presumably from wherever in the cloud it was hiding. If I deleted it again, back it came in even more copies, sort of like the Mickey Mouses' brooms in "The Sorcerors' Apprentice." That fortunately has stopped since Maverick and the last OS upgrade. Fingers crossed it stays that way.
My issue was it's refusing to delete from the iPad. But, doing it properly, left swipe, scroll down to more, then "Trash," does seem to work. Deleting from the laptop also deletes from the phone and the iPad, which is as it should be.
Thanks again.