Bart's got it basically right, except that the horizontal cuts aren't really necessary
One trick - put the onion half flat on the board (Per the photos), and put your "off" hand on top of the onion. Now curl your fingers back so that the last segments are either vertical, or slightly beyond vertical. Now start sliding the onion out from under your hand. The flat of your knife runs along the backs of your fingers. This way you can control how far you move the knife.
Learning how to REALLY use a chef's knife is something all chefs do.
Traditionally, you worked your way up in the kitchen, from plate washer, to pot washer etc. Eventually you got to the point where you start to prep the "Mes en place" (spelling?) Aka, peel the veggies, chop onions, etc - getting everything ready and "In place" (what "mes en place" means)
Mom and Dad started me getting all the veggies and the like ready when I was around 9-10 years old. By the time I was in my mid teens, I did all the basic prep works, so that when Mom got home, she could start dinner. If it was a meal that took a long time to cook, I might even start it (say a roast). The fancy stuff (sauces and the like) Mom or Dad did when they got home. Hey, we got to eat WELL