In IOS mail on iPad or iPhone, are archived emails stored on the device itself or on a server?

Where your Archived mail is stored will largely be determined by your mail settings. Depending upon your email provider, you can generally choose whether to save your Archive mailbox(es) on the mail server, or on your device, for each of your email accounts:

Settings > Mail > Accounts > [Your Mail Account] > [Account] > Advanced > Archive

You will note that the Archive location at the top of the list is “On My iPad”, whereas the bottom entry (selected by default) is “On the Server”

Hi earthlaurie,

We'll be glad to help. This may be dependent on the email service you're using. You can archive mail server side using iCloud, as outlined here: Archive email in Mail on iCloud.com - Apple Support

We hope this clears things up.

Cheers.