When you blink, tears are produced to cleanse and lubricate your eyes. Excess tears are drained away through the tear ducts into the sinus cavity. That is why your nose runs when you cry.
The tears enter the nasal lactermal duct through the punctum. You can see in this diagram that there is a punctual port on both the upper and lower eye lid of each eye.
If your cornea isn't getting enough moisture, you could have dry eye complications. Often times, dry eye conditions can be treated with over-the-counter eye drops. If this type of therapy doesn't work effectively, other treatment options for dry eyes may include prescription eye drops or punctual plug inserts.
Punctual plugs can be used to partially or fully block the tear ducts to help retain tear film moisture. Your eye doctor might insert temporary plugs that may dissolve in a few weeks to test you for compatibility and effectively. If they work well, your doctor can easily insert permanent plugs that do not dissolve.
Talk to your eye care professional for more information about dry eye syndrome and treatments involving eye drop therapy and/or punctual plugs.