Chop the onion in half with all its skin and bits on....
Slice off the longer ends, NOT the hairy ends. You want the hairy ends to hold onto. (Think the bearded bridesmaid in Blackadder.)
Peel the skin off, leaving the hairy ends still on. (You want to get all the brown off - sometimes that involves pulling some white off too.)
Hold the hairy end (not shown - hand holding camera!) and cut lengthways. Don't cut all the way to the hairy end, you want that bit to hold together.
Still using the hairy end to hold onto, cut the other way. When it gets too thin / tricky to cut...
... turn it hairy end up and cut downwards like that.
Until it gets tricky again.
Then just turn it round again.
Voila, a host of chopped onion! (Note the large white bowl in the background. I always have a bowl out to put offcuts into, so I don't have them cluttering up my board or chucked messily on the counter. When the Other One, otherwise an excellent cook, throws peelings directly onto the counter, I have to have a little weep that has nothing to do with onions. Really. It's heartbreaking.)
Final onion thought. For almost every dish that calls for fried onions (I love the thought of dishes calling for this and that, feel free to do the voices), take your time frying these babies. Slow and gentle. Ten minutes, even twenty, hell, why not twenty-five? Watch them soften, brown, sometimes caramelise... At first that might seem a dreadfully schleppy delay, but once you get into it (and have an increasingly well-stocked freezer, so you're only cooking when you want to), you find twenty minutes can be very well used doing other prep, or wandering around the kitchen, or chatting, or what you will. But give them time. They will love you for it.