Because of 1-Sulfinylpropane.
This inconvenient little molecule is produced by enzymes in the onion when you cut it. Normally these enzymes are locked up inside the onion cells, but the knife breaks open the cells and lets the enzymes out, where they make this stuff.
1-Sulfinylpropane is a small molecule, so it can easily evaporate into the air. It gets into your nose and eyes, where the sensors in the eye that protect it from harmful substances get alerted.
To flush out the bad molecules, the eyes and nose produce large amounts of liquid in an attempt to wash away the irritant.
Some onions produce less of this molecule than others. Onions grown in low-sulfur soils generally produce less, and there are sweet onions like the Vidalia that produce less of it.
You can prevent the tears in a number of ways. You can chill the onion in the refrigerator before cutting it. Enzymes work more slowly when they are cold. You can chop the onions under a fan, so the molecule never gets to your eyes and nose. You can chop the onion under water (although just how practical does that sound?). You can wear your protective chemistry goggles, the ones that make a tight seal around your eyes. Or you can use a nice little onion chopping gadget that chops them under a plastic container.