I frequently archive e-mails on my iPhone by mistake. If you accidentally tap the delete or archive button using the Mail app, there is an easy way to bring that e-mail back.
First, you should know that you can set up your mail account(s) to either delete or archive e-mails in the Settings. Deleted messages go to the Trash; archived messages go to the All Mail folder. Either way you set it up, you can retrieve messages that accidentally exit your in-box.
For deleted messages, you will need to find the Trash folder. If you are viewing your Inbox, tap the Mailboxes button in the upper-left corner to return to the main screen of the Mail app. Next, tap your account below the Accounts header and then tap the Trash folder to view all of your deleted messages. Tap on the message you'd like to retrieve, and then tap the folder icon button at the bottom (second from the left). From here, you can tap to move it to your Inbox or any other subfolder you might have created.
The process is the same for archived messages, but instead of digging an e-mail out of the Trash, you'll need to go into the All Mail folder to locate your missing e-mail.