What is the proper way to sharpen a pencil?
Alright, full disclaimer. I wrote that original comment:

I sharpen my drawing pencils the &quot;proper&quot; way, with a knife.

I was intending to be tongue-in-cheek, because you may find a lot of comments or tutorials online insisting that this is the right way to sharpen a pencil.
I don't think that it's the case, but I've now written a little tutorial about the &quot;proper&quot; way to sharpen a pencil, and also some pointers about sharpening pencils in general.
Why use a knife?
Using a knife to sharpen your pencil allows you to control the length and the shape of the exposed core. This, then, gives you more control over the types of strokes you'll be able to make. Longer points mean you can lay down wider lines.
A knife is very helpful for sharpening graphite, charcoal, and pastel pencils. These types of materials hold up to the process better, and wider lines are more likely to be sought from them. As John Cavan pointed out, colored pencils don't really hold up to the process.
What you'll need:


A sandpaper pad or block
A pencil
A very sharp knife
A quick way to get rid of the shavings

The knife needs to be very sharp, because you don't want to apply much pressure to the pencil at all. The more pressure you have to apply, the more likely you are to break the exposed core or cause internal stress fractures that don't show up until much later.
Since the knife is very sharp, please be careful. I prefer to use a utility knife, as shown, because the blades are sharp and easily replaceable when dull. I have used pocket knives before, but sharpening them is a whole other process. If you've never used a utility knife before, be aware that they generally come with spare blades inside them, so you don't have to buy extras right away.
If you've only just started sharpening pencils this way, please use cheap pencils. It's very common to break pencils until you get the hang of this. Practice the technique on something you're fine with throwing away. That said, be aware that some cheaper pencils use really cheap wood, which may be harder or more prone to splitting. You generally won't have that problem with actual art pencils.
I usually sharpen directly over a small trash can, but today I'm doing it over some scrap paper I can just pick up and shake out.
1. Starting
Always hold the knife with the blade facing away from you. If you're concerned about the shavings flying into your eye, wear safety goggles.

Hold the knife at about a 90 degree angle to your pencil, with the blade laying nearly flat against the wood. With the hand holding the pencil, put your thumb against the back of the blade and apply a slight pressure. You will start sharpening by pushing against the knife with your thumb, but holding the knife completely still. The hand with the pencil will do all the work, adjusting the speed and pressure.
Do shallow but long strokes:

You should end up with plenty of shavings, as you can see.
2. Refining the tip
As you start to expose the core, be careful not to nick it. At first you'll start with only a small bit exposed, but then slowly work your way to exposing the full length you want. Make sure you're not just taking wood off the very front, but going back further up the pencil.

As picture above, I started by exposing only a small amount, but then shaved portions away from the entire tip to have more lead exposed. If you only work towards the front, the cone of the wood will be too short and increase the likelihood of breakage.
Once I have what I feel is about the correct length, I test the angle of the wood against a flat surface.

If I can't get the wood flush with the surface, that's no good. The purpose of this type of sharpening is to allow for strokes using the full flat of the lead. If the wood is curved inwards (as seen above) or too short, then trying to use the flat may cause breakage. If the slope doesn't flow well into your actual core, then trying to lay the flat of the pencil down may only yield strokes with part of the tip.
I wasn't happy with my slope, so I worked at it some more. At this point, I do small, quick shavings while rotating the pencil. I'll do this by using the knife to do the work, instead of pushing with my thumb, as it's much faster. However, I've practiced on a bunch of pencils and I'm confident I won't gouge my wood or break the lead.
I ended up with a much smoother slope.

3. Perfecting the point
As you may have noticed, my point is still blunt! You don't need to sharpen that with the knife, as that's where the sanding pad comes in.

Hold the pencil between your thumb and two fingers and roll it so that pencil rotates. You'll do this while rubbing the lead point along the sandpaper. Make steady, even strokes. Like before, only the pencil should move, and not the sandpaper. As you roll the pencil, the goal is to get a full 360 degree rotation, or close, in one full stroke.
The reason for this rotation is that it keeps the point centered in the core and keeps it conical. If you don't rotate enough, you may end up with more of a wedge or oblong shape. Those can be quite useful, if it's your goal, but generally you don't want those other shapes.
You can also sharpen the point by sanding with your strokes following the length of the pencil, instead of perpendicular to the pencil. (I think of it as &quot;up and down&quot; instead of &quot;side to side&quot;).

This lets you more easily match the point to the slope of the wood, if that's you're intent. You still need to rotate the pencil as you do this. I spent time making my slope nice, so I used a combination of strokes to get the point to match.
In the end, I made a nice, long point.

As you practice, you may end up making short or longer points, depending on your preference (or skill). The longer the point, the wider the strokes you can make, but also the more likely you are to have breaks while sharpening, using or storing your pencils.
4. Clean up
Your sandpaper pad may look pretty grim after this. But if you bang it on your table or inside your bin, a fair amount of dust will come off. You can use one sheet for a considerable number of pencils, so don't throw it away too soon.

Before

After
Then, before actually using the pencil, you need to clean it off, too. I typically make a bunch of test strokes, and even rub just the wooden part on the paper. A lot of the dust from your core will laying on the surface, and if you don't wipe it off you may end up laying down some accidental, very dark lines.
You can use a cloth for this, but I usually just use scratch paper.

The proper way, in general
The keys to a sharp pencil, without breakage, are generally the same whether you're using a knife or a hand sharpener:

Have a sharp blade
Move the pencil, not the blade

I have a handful of sharpeners just like this:

They work great for me. I just keep it steady in my offhand, and turn the pencil with my other. The blades are replaceable, but I have so many sharpeners that I haven't had to do that, yet.
In fact, I only sharpen one set of by 9B - 9H range of pencils with a knife, and the rest get the hand sharpener treatment.
I avoid electric sharpeners for anything but writing pencils. They violate one of my guidelines, because the blade moves and not the pencil, so I already distrust them. Something about the process seems to increase the chance that the lead cracks internally, and I just end up sharpening them over and over.
They also get gummed up by colored pencils. Some companies make electric sharpeners just for colored pencils, but I haven't tried them so I can't comment.