Gmail - Message Management - Labels and Deleting Messages

Gmail doesn't have folders.  All your messages are saved in All Mail.  Everything else (Inbox, Sent Mail, Starred, labels, etc) are just "views" into a sub-set of the messages in All Mail.  And since there's only one copy of any message, if you delete it from any label, you're deleting the only copy, so it's gone from all of them.  If you delete a label you remove that "view", but the messages are still in All Mail.

When you Archive a message, the only thing that does is remove the Inbox label from the message.  The original message is still in All Mail along with all your other messages, and you can apply labels to them to make them easier to find later.  Remember that Archive is an action, not a location.

Labels actually provide a super-set of functionality compared to traditional folders.  You can do everything you can with folders, and a number of things that folder do not support.  The main difference is that there are not multiple copies of messages when multiple labels are applied so (as described above) deleting a message from one label deletes the only copy so it's deleted from all labels.

For more information see:
Using labels:
Labels vs Folders:
Archiving mail: