I learned this watching the Food Network years ago and it changed my life. I've cut a lot of onions, knowing this trick would have been awesome back when I worked at Zesto's as a kid.
Start by cutting your onion in half.
Not like this:
Notice that the knife below is positioned to slice through the bottom root section and the top stalk section simultaneously:
Peel and cut off a small portion of the top to remove any dried portion of stalk. Do not cut the root side:
Now slice the onion without cutting all the way to the root section. 2 photo's below:
No turn onion and slice perpendicular to your previous cuts:
Creating small even pieces thanks to the layers already in the onion:
Like my daughter says "Easy Peazy Lemon Squeezy"
I need to apologize I have had many people read this hoping to find out how not to cry while cutting an onion. It has been so long since I've had that problem I really had forgotten that it was even a problem! I store my onions in the fridge, the cold inhibits the release of the eye-irritating juices. (FYI - Onions may cause your other veggies in the fridge to spoil quicker though). It also probably helps that I chop quick using the above method so there isn't much time for crying, and I have the wingspan of an eagle so the onion is a pretty good distance from my eyes. Happy Chopping!