Chopping an onion the right way makes it quick and gives a nice consistent chop. Using this technique and given a sharp knife, I can usually get an onion chopped and in the pan before my eyes start to water. Also note that this technique, given practice and skill, can be used for chopping shallots and garlic.
  1. Take your unpeeled onion and cut it in half.
  2. Take one half and peel it, now that you have nice edges to grab. Cut the roots off from the root end.
  3. Make a series of partial cuts towards the root end, but not separating the onion from its whole, cut all the way to the table.
  4. Make 2 or 3 horizontal cuts parallel to the table - be careful here as you're bringing the knife towards your hand in part of this motion.
  5. Complete the chopping by making a final series of normal vertical cuts, with the onion rotated 90 degrees from step 3.

Voilà - see the either cool or 'good intentions gone horribly wrong' ascii images below.

     , \ | ,'
      i-\':_-
   ,-'   |  `-.
  /      |     \
 /       |      \
|        |       |
|        |       |
 \       |      .'
  \      |     ,'
   `.    |   _,
     `-..|.-'
         |
   1. Cut Here
       .  |  ,'
        `.|,'
 ....................
    /           \  Cut
   /             \
  |               |
  |               |
   \             /
    \           /
     `-._   _,-'
         `''
  2. Take one half, peel
     it and cut roots
    ,'---------.
   /  . . | .   \
  / | ' | ' | |  \
  | | | | | | | ||
  \ | | | | | | ||
   \| | | | | | |'
    '.| | | | |/
      `.|_|_'-'
 3. Make a series of
    vertical cuts
              _____
      --->_.-'     `-,_
    --> .'             `|
       /                |
 ---> .`                |
     .|.................|.........
       4.Make 2 or 3             Table
        horizontal cuts

     |,|-+--|.|
   ,'|.|.|..|.|-|
  | -+-+-+--+-+-+\|
  | .|.|.|..|.|.|.|
  |  |.|.|..|.|.|.||
  |  +-+-+--+-+-+-+|
  | .|.|.|..|.|.|.|
  +  |.|.|..|.|.|'|
   `.| |.|..|.|'|
     |`|-+--| |
   5. Make a last set
     of vertical cuts
     (rotated 90 deg)

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