Using a knife, cut a cone out of the bottom of the onion (where the roots come out). The diameter of this cone should be about a third of the diameter of the onion, and about 1/3 deep. Take this piece and throw it away (don't put it down the disposal!). This piece contains the part/gland that makes baby Jesus and everyone else in the room cry when you're chopping it up. Once you've gotten that piece out, chop off the top, peel, and slice the onion.
Refrigerate the onion 30 minutes prior to chopping.
When chopping, don't cut the root off. The root contains the most of the chemical that causes eye irritation.