Inside the onion cells there are some chemical compounds which contain sulfur. When you cut an onion its cells are broken and those chemical compounds then undergo a reaction that transforms them into a more volatile sulfured products, which are released into the air.
These sulfured compounds react with the moisture in your eyes forming sulfuric acid, which produces a burning sensation. The nerve endings in your eyes are very sensitive and so they pick up on this irritation. The brain reacts by telling your tear ducts to produce more water, to dilute the irritating acid. So you cry to keep your eyes protected from the acid.
There are some tricks to make onion-dicing less problematic:
Chop the onion under cold water. The volatile sulfured compounds will be released but then they react with the water, instead of reaching your eyes.
You can freeze the onion for 10 minutes before cutting it. The cold temperature of the onion will slow down the chemical reaction which forms the volatile sulfured compunds.
