You can cut the onion in half and let it rest for about 15 minutes in the fridge under a piece of saran (Don't know if this is the correct word but it's a thin, quite sticky sheet). Normally, you shouldn't have those tearing eyes anymore.

I wear contact lenses, and they are the BEST for preventing onion tears. Not that anyone would get contacts for this purpose, of course, but it's a nice side benefit. When I wear my glasses, they offer some protection, too.

I saw this method on a TV show, and it's about 90% successful.

DO NOT RUB EYES at any stage.

1. Cut the 'flower' end off (NOT the root).
2. Peel the outer leaves off to your desired depth.
3. Place onion down onto the area you cut off in 1 above.
4. Cut onion in half through root.
5. Chop and slice one half of your onion as you need to leaving root intact.
6. Repeat with other half.
7. Repeat to remaining onions until all are cut, chopped or diced.

If you have a gas stove, turn on a burner and chop your onions next to the flame.

You don't have to buy specialty goggles. Ski goggles work really well. You can pick up a pair of science/laboratory goggles. I would think that these would be cheaper than something labeled as a "Tear-free Onion Cutting Goggle ".

recent pickle on the topic:

http://www.food52.com/foodpickle/2916-what-s-the-best-way-to-cut

American's Test Kitchen tried all of the above mentioned and the only thing that really worked was the onion goggles. You can find them at many places. The most recent place I saw them was in a King Arthur Catalog.

You can't stop the gasses that release when you cut an onion, but you can do things to help. Alton Brown wears goggles, iirc. I find chilling the onion helps. I have heard that bread in mouth helps but haven't tried it. Use a sharp knife to make clean cuts helps.

* hold a piece of bread in your mouth while cutting the onion
* wear goggles