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Slicing an onion ruptures its cells and sets loose chemicals that drift up into the air around your face.
Cutting an onion ruptures its cells and releases enzymes and sulfur-containing compounds that were previously kept separate inside different cell compartments. These substances mix and react in the air, forming a volatile gas that drifts upward toward your face. When that gas reaches the surface of your eyes, it dissolves in the tear film and forms a mild acid, irritating the eye, so your body responds by producing more tears to flush it away.
Onions are not unusually cold inside, so temperature has nothing to do with the stinging sensation, and while the smell can be pungent, it is the chemical reaction in your eyes, not any actual toxicity, that causes discomfort.
Chilling an onion before cutting, or cutting it under running water or near a fan, slows the release and dispersal of the irritating gas, which is why so many kitchen tricks for tear-free chopping focus on temperature and airflow.
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