This checkbox (in System Preferences > Displays) lets you toggle between mirrored displays and dual displays.

I have added presenter notes to one of my Keynote presentations (by choosing View > Show Presenter Notes), but whenever I run the program in Rehearsal mode (View > Rehearse Slideshow), it shows my notes on the projector screen. I can’t figure out how to show my notes on my personal screen but not the public one.

Rehearsal mode will always show your presenter notes on both screens, because that mode is designed for when you don’t have an external projector connected to your Mac. When you do have a projector connected to your Mac, simply play your slideshow normally—either click the Play button in the upper-left corner of the screen, or choose View > Play Slideshow.

If you’re not seeing your notes on either screen, be sure that Keynote’s preferences are set to show your presenter notes, because they aren’t selected by default. Go into Keynote’s preferences, click the Presenter Display button, and check the Notes checkbox.

If Keynote shows your presenter notes on both screens, then your computer is set to mirrored displays instead of dual displays. You can toggle these display modes while your projector is connected to your Mac by going into your System Preferences, choosing Display, clicking the Arrangement tab, and deselecting the checkbox that says Mirror Displays.