What do white rings around the eyeballs indicate?

Could be normal: Hard to tell if this is what you're describing, but as one ages one can develop a slight light colored ring around the iris (colored part of the eye). This is called arcus senilis and is benign.

High cholesterol: White rings around the colored part of the eye are cholesterol deposits and often are a sign of high cholesterol. Make sure your cholesterol is under control.

Arcus Senilis: Arcus senilis is the technical term for the whitish or bluish rings around the cornea of the eye in people generally above 40s or 50s. It is usually due to deposition of cholesterol and might indicate heart disease (atherosclerosis) it has nothing to do with visual acuity or vision in general. Best to get a physical exam and also see optometrist or opthalmologist.