Cut the onion in half through the roots and top stem.
Place one half flat on the chopping board. Cut off the stem and discard, but leave the root intact.
Turn the onion so the root is facing the back of the chopping board. Cut vertically up to the root. (Be careful not to cut all the way through the root or the cut won't work.) The thicker the slices, the bigger the dice.
Turn the onion so that the root is on the opposite side of the knife-hand. Hold the knife parallel to the chopping board (180 degrees) and make horizontal cuts up to the root. Again, be careful not to cut all the way through. Depending on the size of the onion, about 2-3 slices will suffice.
Return the knife to its regular cutting position (90 degrees) and slice the onion in the opposite direction of the first cuts. Slice until you reach the root and discard it.
To make it even finer, scoop the finely diced onion into a pile and continue to chop until it reaches your desired size.
Wedges (Kushi-Giri)
Cut the onion in half from the root to the top.
Place it flat on the chopping board, cut off the top and discard.
Cut a V shape around the root and remove without separating the layers. Discard the root.
Cut the half in half.
Cut the quarters in half again, or angle your knife to create different thicknesses according to your preference.
Round Slices (Wa-Giri)
Take a whole onion, cut off the top and discard.
Hold the onion firmly (be careful of slipping) and slice to your preferred thickness.
Thinly Sliced (Usu-Giri)
Cut the onion in half from root to top.
Cut off the top and discard.
Slice to your desired thickness until you reach the root.
Throw the root away and repeat with the other half.