I learned this technique for slicing apples some years ago when my mother went back to school to become a pastry chef. This is the only way I core and slice apples now, because it really is so superior to the old fashioned way of quartering an apple and using a paring knife to cut a wedge-shaped section out of the middle to remove the piece of apple core.
Start by imagining a square around the core of the apple if you're looking down at top of the apple. You want to make your first cut just to the outside of the apple core.
Slice off the first lobe of apple, and you're left with a nice, flat, steady surface. Turn that flat, cut side of the apple face-down on the cutting board.
How the apple is very secure, and is not in danger of rolling around as you cut the next slice. Again, cut just to the side of the apple core.
Continue slicing to the side of the apple core, making four cuts to remove all of the delicious apple fruit from around the center core.
There's the tiniest bit more waste when cutting an apple with this method, but it's not much of a difference at all, and the improved safety and speed makes this technique worth it.
The pieces of apple you're left with, since they also have a nice, flat, steady side, are much safer to cut into slices than standard wedges of apples.
Thanks for teaching me your pastry school tips and tricks, mom!