Apple - iOS 7.1 - Where do emails go to when I 'archive' them in Gmail App?

When you archive things in gmail, it just removes the 'inbox' label on it. If you want to find the archived email, just search for them or you can click on the label 'all mail' on the left. The same goes with gmail online. see more at https://support.google.com/mail/answer/6576?hl=en

If you really want to delete the emails, you can just hold your finger on the "Archive" button for a second or two, and choose "Trash selected messages".

@shub's answer is correct as per the original question, though. (SO won't let me comment yet, or this would just be a comment.)