Often minor modes don’t respect standard keybinding conventions and use some user reserved keys (like C-c a). Occasionally two minor modes would have conflicting keybindings or a minor mode would keybindings conflicting with a major mode. I guess you can imagine similar problems. Dealing with them is fairly straight-forward - we have to either unbind or rebind the offending binding:

;; remove keybinding
(define-key minor-mode-map (kbd "C-c c") nil)

;; change keybinding
(define-key minor-mode-map (kbd "C-c c") (kbd "C-c C-c"))

Generally speaking you can use define-key to alter the keymap of a major mode as well, but those are rarely offenders when it comes to picking appropriate keybindings for their keymaps.

Normally you’d want to invoke the above code right after the related minor (or major) mode is loaded:

(eval-after-load "minor-mode"
  '(define-key minor-mode-map (kbd "C-c c") nil))

Making a minor mode have different keymaps in different major modes is tricky, but possible. Here’s an example that disables some keybindings in the minor prelude-mode, that are conflicting with the major org-mode:

(defun my-org-mode-hook ()
  (let ((oldmap (cdr (assoc 'prelude-mode minor-mode-map-alist)))
        (newmap (make-sparse-keymap)))
    (set-keymap-parent newmap oldmap)
    (define-key newmap (kbd "C-c +") nil)
    (define-key newmap (kbd "C-c -") nil)
    (make-local-variable 'minor-mode-overriding-map-alist)
    (push `(prelude-mode . ,newmap) minor-mode-overriding-map-alist))

(add-hook 'org-mode-hook 'my-org-mode-hook)