If you're using an email service that supports it (such as an iCloud account or a Gmail one), you can choose to archive your messages instead of deleting them, which will save emails into a folder called "Archive" or, within Gmail, "All Mail." If this feature is turned on for a particular email account on your iOS device, you'll see a storage box in your toolbar when you've got a message open instead of the typical trash icon.
Additionally, if you swipe across a message to get rid of it, the red button will read "Archive" instead of "Delete."
Now, you can turn this off for each of your accounts if you want to. Just go to Settings > Mail, Contacts, Calendars, pick the proper account, and toggle the Archive Messages option off.
If you do that, your device'll put removed emails into your trash instead.
But what if you want to archive certain messages and delete others? Well, you could open your Archive/All Mail folder and delete the messages from there. For a faster way, though, first make sure the archiving option is on for the account in question. Then just press and hold briefly on the file folder icon when you've got a message open, and you'll be given the choice of what to do with it.
Hooray! So some emails can go to your archive, and others can go to the trash. Maybe in your case, though, this tip was about how to turn the infernal archiving feature off. Whatever floats your boat, blows your skirt up, or…um…puts pajamas on your cat, or something.