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Tap your Android's Gmail app to open Gmail. Open the Gmail Menu. Tap "Settings". Tap "General Settings". Tap "Gmail Default Action". Tap the "Archive" option. Return to your inbox. Search for an e-mail to archive. Select a message. Tap the Archive button. Access your archived mail in the "All Mail" folder.
To archive mail on Android, you'll first have to switch your default mail selection settings from "Delete" to "Archive". While you can technically archive individual messages from within the email itself, doing this is not conducive to mass-archiving. You can do this by tapping the three horizontal lines in the top left corner of your app. This will open an extended list of your Gmail app's settings. This will open another settings menu. You can change your default selection option--archiving versus deleting--from this menu. This will make archiving your default selection option. You can also choose to receive a confirmation notification before archiving/deleting emails from this menu. You'll need to select the emails you wish to archive from here. If you have keywords or subject matter for which you want to search, you can do this from the search bar at the top of your screen. Do this by tapping the checkbox to the left of an individual email, then tapping subsequent emails to select them. This is the downward-facing arrow at the top of your screen; doing so will remove your messages from the inbox and place them in the All Mail folder! You can also archive individual emails by swiping left over the email. You can do this from within the Gmail menu; note that you may have to tap "Show All Labels" to display the All Mail folder.