If a file is set to read only mode, how do I change it to write mode and vice versa from within Emacs?
As mentioned up there by somebody else: M-x toggle-read-only would work.
However, this is now deprecated and M-x read-only-mode is the current way to do it, that it is set to C-x C-q keybinding.
If only the buffer (and not the file) is read-only, you can use toggle-read-only, which is usually bound to C-x C-q.
If the file itself is read-only, however, you may find the following function useful:
(defun set-buffer-file-writable ()
  "Make the file shown in the current buffer writable.
Make the buffer writable as well."
  (interactive)
  (unix-output "chmod" "+w" (buffer-file-name))
  (toggle-read-only nil)
  (message (trim-right '(?\n) (unix-output "ls" "-l" (buffer-file-name)))))
The function depends on unix-output and trim-right:
(defun unix-output (command &rest args)
  "Run a unix command and, if it returns 0, return the output as a string.
Otherwise, signal an error.  The error message is the first line of the output."
  (let ((output-buffer (generate-new-buffer "*stdout*")))
    (unwind-protect
     (let ((return-value (apply 'call-process command nil
                    output-buffer nil args)))
       (set-buffer output-buffer)
       (save-excursion 
         (unless (= return-value 0)
           (goto-char (point-min))
           (end-of-line)
           (if (= (point-min) (point))
           (error "Command failed: %s%s" command
              (with-output-to-string
                  (dolist (arg args)
                (princ " ")
                (princ arg))))
           (error "%s" (buffer-substring-no-properties (point-min) 
                                   (point)))))
         (buffer-substring-no-properties (point-min) (point-max))))
      (kill-buffer output-buffer))))
(defun trim-right (bag string &optional start end)
  (setq bag (if (eq bag t) '(?\  ?\n ?\t ?\v ?\r ?\f) bag)
    start (or start 0)
    end (or end (length string)))
  (while (and (> end 0)
          (member (aref string (1- end)) bag))
    (decf end))
  (substring string start end))
Place the functions in your ~/.emacs.el, evaluate them (or restart emacs).  You can then make the file in the current buffer writable with M-x set-buffer-file-writable.
If you are looking at a directory of files (dired), then you can use Shift + M on a filename and enter the modespec, the same attributes used in the chmod command.
M modespec <RET>
See the other useful commands on files in a directory listing at http://www.gnu.org/s/libtool/manual/emacs/Operating-on-Files.html