I have small little zits or bumps under my eyes, they seem to be clogged pores or something. What causes this and how can I get rid of them?

There are two primary possibilities here other than acne. The first are called milia. These are small collections of skin protein (keratin) that becomes trapped under the skin. This can either be an inherited problem or it can result from damage to the skin over time from the sun and other factors. Unfortunately, milia are closed off and do not open to the outside of the skin through a pore, therefore they cannot be squeezed out or exfoliated away like can be done with acne. Another possibility would be syringomas. These are small benign tumors of the sweat glands under the skin that typically collect in the area under the eyes. Neither milia or syringomas require treatment unless they are cosmetically bothersome to you. If they are bothersome, then you will need to talk to your dermatologist, as there is no effective home remedy for either condition. Treatment of milia may be as simple as having them punctured and drained by your doctor if they are few. However, if they are extensive they may need a dermal abrasion or laser based treatment. Syringomas can similarly be treated with abrasion or laser based treatments. Talk to your dermatologist to figure out what your options are.