This fix is super easy and you'll be amazed at how well it works. First of all don't panic and don't rub the spot, but you will need to try to work quickly before it gets too dry. Begin by taking several dry paper towels and gently press the spots in order to lift the majority of the excess nail polish.
Next spray the spot with a regular glass cleaner (Windex or whatever brand you have on hand) and then at first blot with clean dry paper towels. Once you lift most of the nail polish you can then spray the spot once again with the glass cleaner and gently rub the carpet quickly in a circular motion, changing paper towels until the spot is gone.
I had to remove half a bottle of dark glittery blue nail polish from a lighter gray carpet, and the glass cleaner worked amazingly. The lighter the carpet the harder it will be to remove of course, but with a little elbow grease it should work with no problem.