A significant numbe of people make a real mess of chopping onions. Im pretty sure its somthing I learned watching Ready Steady Cook in about 1995.

Cut the top off the onion

cut the onion in half lengthways (through the root).

peel the obion, leaving the root intact

Slice the onion lenthways BUT do not cut all the way to the root

Slice the onion crossways, leaving little tiny pieces of onion. Throw the root way.