When you cut an onion, you break the cell walls, allowing different chemicals to mix and react with each other. In particular, enzymes called alliinases break down the compound allicin into a very pungent sulfenic acid which diffuses through the air, burning your eyes and nose. This reaction probably helps keep the plant alive, since it will irritate most animals who would otherwise eat it. You can avoid this reaction by cutting the onions under running water or even underwater.

Onions contain a poison that is released into the air when they are cut open and causes eye watering in humans. In smaller amount, we enjoy the taste of this toxin. It is more toxic to some other animals, despite being fairly benign to humans.

There probably are multiple versions of this toxin, or multiple toxins. Leeks and garlic are basically different kinds of onions, but they taste different enough (due to said chemicals) that we call them by different names. I don't know whether the types of poisons they contain are different, or if they contain the same things but in different amounts.