Dicing an onion is only a little more difficult than dicing a carrot. First step is to cut the onion in half through the root end. Be sure to leave the root intact when you peel off the skin because you'll need the root to hold the onion together.
To dice the onion, hold your knife parallel to the cutting surface and carefully slice the onion, stopping short of the root. Make each slice the width of the final dice.
Next, make vertical slices, each the width of the final dice. Be sure not to cut through the root.
To finish the dice, slice across the other two cuts.
With a bit of practice you'll be able to finish dicing an onion before it has the chance to make you cry.