Cut onions under cold running water to help redirect the vapors away from your eyes.
Breath through my mouth so that the vapors are absorbed through your mouth before they reach your eyes.
Sharpen your knife as a sharp knife will cause less fumes to be released from the onion.
Place the onions in the freezer for 15 minutes or soak them in water for 15 minutes before cutting them as chilled onions release less compounds and the water will absorb the compounds that cause the eye irritation
Cut the onions around a fan or a vent hood so that the irrants are pulled away from your eyes.
Make sure you cut the root edge off last as many of the fumes are held in the root.
If all else fails, wear googles while you cut onions to help protect your eyes from the fumes.