you cry because Onions produce the chemical irritant known as syn-propanethial-S-oxide. It stimulates the eyes' lachrymal glands so they release tears. Scientists used to blame the enzyme allinase for the instability of substances in a cut onion.
I heard that to prevent you from crying Chop the onion under water, under running water, or pre-soak in water.
This 'under water' lore seems quite pervasive; I cannot imagine trying to hold and contain all the diced onion bits (which float away) under a sink filled with water. In theory if the onion was exposed only to water and not air the sulfoxide would be washed away by the water and never become airborne. A variant is to have running water nearby the chopping area in the hopes that it will obtain some of this aforementioned magical attractive power (this didn't work for me.) Another scheme is to soak the onion peeled (even cut in half or quarters) in water for some length of time to draw out the juices. While this may reduce the fumes some it only penetrates so far; once I started slicing into onion the spray was as bad as before.
Wear contact lenses to shield eye surface from airborne spray.
I don't wear contacts so I can't say firsthand, but they don't cover the whole eye surface (leaving it partially exposed) and don't do anything for the nose. Besides, see the next suggestion.
Don't wear contact lenses.
The acid gets between the lens and the eye for an even more excruciating pain.
Wear swim goggles or a diving mask.
This solution actually works because the particles in the air are prevented from touching your skin. The goggles will only protect your eyes, and with the mask you have to breath out of your mouth. It is cumbersome and uncomfortable, and it doesn't protect any one else in the room.
Use a fan to blow away or suck up the fumes.
If you can get a strong enough fan or cut your onions outside where there is a breeze, I can see this method as somewhat effective. If inside, wherever you blow the fumes (unless it's out a window) will be fumigated. Some say use the fan over the stove top, but it usually isn't strong enough; besides it draws the air up, where your head is likely to be. With a fan it is best to blow horizontally, away from the face.
Place onions in the freezer for 20 minutes before cutting.
When the onion juices are chilled they are less likely to become airborne. This strategy is the most well-known solution to the onion problem, but it simply does not work. Unless the onion is actually frozen, there will still be onion juice sprayed in the air (besides, unless you store them in the refrigerator you have to wait for the onion to chill before you can cut it.)
Put white vinegar on the chopping block to neutralize chemicals.
I tried this method, and it actually works. The stinging in the eyes was almost eliminated, but the smell was not very pleasant. The moist vinegar on the onion interferes with frying, and its flavor may taint some foods.
Burn a candle near the work area.
This rarely-known technique is based on the fact that the candle flame pulls air from around it and sends it up the smoke plume. Although surprisingly effective, a single candle usually cannot handle the large amount of aerosol generated by a lot of dicing or fine slicing. This last technique holds the secret to the final solution.
And now the answer you've all been waiting for:
Place the cutting board next to a gas range and turn on a burner or two.
The heat and flames draw in the onion mist, burn it, and send it up with the rest of the flame exhaust. In the heat the chemical changes such that it no longer irritates the eyes. One could also use the hood exhaust fan while the burners are on. The only drawback with this solution is that some unlucky fools aren't cooking with gas. They could try an electric range on high, but it doesn't really work (better to use several candles instead.) A poor stove arrangement might make it difficult or force you to cut left handed, but it can usually be worked out (and you don't have to stand very close to the flame). As with the fan, you want the flame to pull the onion fumes away from your face. And as pertains to all methods, you want to keep your face as far away from the hacking and hewing action at the cutting board as you can. I have a range with four burners: two on the left and two on the right with a big space between them where I put the cutting board. I turn on the two right burners, stand a bit to the left, and cut with my right hand.