When sharpening, there is an issue of bevels. Your fancy knife has an edge that consists of one or more of these guys, so some questions: how do I create them and what should they look like?
For single beveled knives, the typical approach is to flatten the secondary bevel and cut a primary “micro” bevel into it. The micro bevel is what you actually cut food with, that secondary bevel is pretty large and goes up about halfway up the blade. When you sharpen a knife like this, you grind down the secondary to flat and grind out the primary bevel in the process. When finished, you re-cut the primary.
Western or double-beveled knives (sharpened on both sides) are generally much thinner spines than single-beveled knives. As such, there isn’t a lot of knife to cut two bevels into — so most forgo the secondary and simply cut the primary directly into the blade.
I lifted these images off of the Knifeforums. Click through for the blowups ….

Single beveled knife with a micro bevel
Another wrinkle — hamagurabi edges!
“WTF is that?!?” I hear you ask. Excellent question.
In English, hamagurabi, means “clam shell”. In this context, we really mean convex. So, no sharp bevel transitions — smooth those out a bit. If you’re sharpening freehand, the slop in your strokes will probably do this for you “for free”, but if you’re using a device like a Spyderco, Apex, or Gizmo, the bevels are going to come out a whole lot more precise, with abrubt transitions. Is this bad? No, not really, but some argue that a clamshell bevel will be stronger, will wear better, and will cut better.

Single beveled edge with hamaguriba sharpening
Here’s the diagram with the “rounding” made a bit more obvious:

Thanks to Thomas/Throw3636 for posting the original diagrams online for us all.
Some videos (in Engrish) for those interested in following along: