What does it mean when the whites of someone's eyes have a discoloration (not talking about pink-eye)?

Several things. Depending on the color, shape, size and location, there are several possible causes. Most commonly, yellow "fatty" appearing spots near the cornea (clear part covering the color of the eye) called pingueculae or pterygiae are caused by years of exposure to the elements (sunlight, high winds). Any newly-seen spots should be evaluated by an eye doctor.
Many things. Divide discoloration into red and non-red. Red usually indicates inflammation (chemical, allergic, infectious, autoimmune) or trauma(bright red subconjunctival hemorrhage). Non-red discoloration would include congenital or acquired melanin deposits, pinguecula (yellowish bumps from sun/dry air exposure, yellow from liver failure (jaundice), black from silver nitrate exposure in addition to more.