Why do my eyes turn red and watery when I use makeup or eye shadow?

Possibilities. You might be having an irritant reaction or you could be having an allergic reaction. Visit an allergist in your area to discuss this issue. They might recommend patch testing to identify if you have a sensitivity to an ingredient in the makeup.
Allergy or Irritant? This could represent a contact allergic reaction due to preservative, and fragrances in the cosmetics. Even "hypoallergenic" products can cause a reaction, especially if your tissues are already irritated. It could also just be an irritant reaction due to a foreign product near the eyes. Either way i would discontinue the products and let the area return to normal.