How to remove all files ending with ~ made by Emacs

While all the others answers here correctly explain how to remove the files, you ought to understand what’s going on. Those files ending in ~ are backup files, automatically created by Emacs. They can be useful sometimes. If you’re annoyed by the files and want to delete them every time, then you either

(1). prevent the creation of backup files:

(setq make-backup-files nil)

or

(2). Have it save the backup files in some other directory, where they won’t bother you unless you go looking for them. I have the following in my .emacs:

(setq backup-directory-alist '(("." . "~/.emacs.d/backup"))
  backup-by-copying t    ; Don't delink hardlinks
  version-control t      ; Use version numbers on backups
  delete-old-versions t  ; Automatically delete excess backups
  kept-new-versions 20   ; how many of the newest versions to keep
  kept-old-versions 5    ; and how many of the old
  )

(Only the first line is crucial.) To see documentation about backup-directory-alist, type C-h v backup-directory-alist.