Deleting and archiving messages on an iPhone or iPod touch works differently depending on your device's software version, or how you it set up.
If you're running version 1.1.2 OR you've manually set up IMAP by following these instructions, this is how message deletion and archiving will work:
Tap the trash icon while reading any message > Archives message so that it will only appear in [Gmail]/All Mail and your iPhone 'Trash' folder.
Note: This is how Gmail IMAP works in most mail clients. Learn more
If you're running version 1.1.3 or later AND you've automatically configured Gmail by tapping the large 'Gmail' logo during setup, the same actions will differ in the following way:
Tap the trash icon while reading any message > Moves message to the [Gmail]/Trash folder, where it will be permanently removed in 30 days.
Important: If a message has any other labels, the message will also be deleted from its corresponding IMAP folders.
If you wish to archive a message in this new implementation, simply move the message to your [Gmail]/All Mail folder. This can be done by opening the message on your iPhone, tapping the folder icon located between the trash and refresh icons, and then tapping your [Gmail]/All Mail folder to move the message.
For additional information regarding this behavior, please visit Apple Support.