Those are special folders, which cannot be deleted or renamed. They correspond to the folders ("labels") you see on the Gmail web site.
If you want to make the Gmail server actually delete something, move it to "Trash" from "All Mail"
The main advantage of Gmail is lost with any IMAP / POP / Exchange email client:
Labels instead of folders.
A new message appears simultaneously in "All Mail" and "Inbox." A message or conversation thread can exist simultaneously in multiple labels.
When you use the "archive" function in the Gmail app or web UI, it simply removes the Inbox label. The message or conversation still exists in "All Mail" and any other labels you have applied.