In emacs, my <C-right> and <C-left> on my mac keyboard, does not work

Actually, I found that ctrl-left arrow and ctrl-right arrow did nothing, even for other applications on my Mac.

Under Keyboard -> Keyboard Shortcuts in System Settings, you'll have to deselect the Mission Control shortcuts, which appear to have been added through some OSX update.

⌃← and ⌃→ worked for me by default in Emacs.app and when I ran emacs from iTerm 2, but not when I ran emacs from Terminal.

If you use emacs in Terminal, try changing ⌃← from \033[5D to \033[1;5D and ⌃→ from \033[5C to \033[1;5C. You can insert \033 by pressing the escape key. \033[1;5D and \033[1;5C are the default values of the keys in iTerm 2.

If ⌃← and ⌃→ don't work in Emacs.app either, the problem might be that they are used for changing desktops, but you only have one desktop open, so they don't seem to do anything. Check System Preferences > Keyboard > Keyboard Shortcuts > Mission Control.

If you also need C-up and C-down in Terminal.app for Emacs, try adding the following keys under Terminal > Preferences > Profile > some profile > Keyboard:

^↑  \033[1;5A
^↓  \033[1;5B