We use them for everything, but chopping onions can be annoying. I say annoying not only because they make us cry (if you have contacts, you're lucky! You have less surface area exposed to the irritating compound) but also because their layered structure easily falls apart.
Today's tip is two-part:
1) Use a sharp knife- the less your onion gets pushed around, meaning the better your knife slices right through it, the better the onion will hold its shape.
2) Keep the root of the onion attached; don't slice the root end off. Peel the onion, slice the shoots coming out the top, and then cut it lengthwise through the root. This cut will leave you with two halves of an onion, its layers held together by the still-attached root. Make your vertical cuts with the grain, careful not to cut all the way through the root end, and then turn 90 degrees and make your little squares. When you get to the root end, your last cut will free the root and then you can toss it, just like you would have earlier.
The onion wants to hold itself together; I just had been cutting off the root prematurely and thereby making my life harder.
Now if only I could figure out a way to stop crying...