There are many ways to sharpen a knife. This method produces a good general purpose edge.

Safety note: As my Granddad used to say: "Don't cut toward yourself, and you'll never get cut."

Excellent photos by Christy Canida the whale butcher.
See what she does with these knives at Skinning and Filleting Catfish

People have always cared about sharp tools. Some "Bog Man" remains from thousands of years ago have been found with sharpening stones worn as a pendant.

This first video shows how to make your own Bog Man stone from a regular sharpening stone, or any soft abrasive stone you happen to find.

First we will make a drill bit from a nail, drill a hole in the stone, saw the stone in half, and flatten it. Just like an ancient bog man would have if he had the battery drill we fixed last week.

This second video shows how to sharpen a knife for butchering and how to sharpen it for carving wood. I bought the knife in the video from a husband-and-wife team of blacksmiths in China. Blacksmithing seems to be a job for couples in many parts of the world.

Links to the ipod formated videos are at the bottom of this page.

Step 1: Look at the Edge

Get under a bright light such as the sun, and hold up the edge. You'll see reflections on flat spots and nicks.
On this blade the inch near the tip is pretty bad.

Step 2: Thumbnail Test the Edge

Touch it to your thumbnail and see if it slides around or if it catches.
If it slides that means it's dull, as in not sharp, at least in this area.
Safety note: Don't chop your fricking thumb off.

Step 3: Flatten the Stone

Buy yourself a sharpening stone for a dollar in Chinatown. If your city doesn't have a Chinatown, get one or move somewhere civilized.

This is a "water stone" which means you put water on it while using it to float the sharpening dust off it. Some prefer an "oil stone" which means you put oil on it. Some stones are born oily. Once the oil is in there water doesn't work well anymore.

This particular stone has had some use so it's dished out in the middle. That makes it hard to hold the knife at the right angle. So we'll need to flatten it.
Actually it doesn't really matter for knives, but when you start sharpening plane blades you'll make a religion of flattening your stone.

Step 4: Flatten Away

Splash some water on the sidewalk and rub the stone on it til the stone is flat. Use plenty of pressure.
Listen to "The Great War for Civilization" by Robert Fisk while you work to understand what went wrong in the Middle East.

Step 5: Thin the Edge

Rub the knife on the coarse side of the stone at a 5 degree angle as shown. It doesn't matter what stroke you use or what direction.
You're just thinning the area around the edge a bit to save you some labor later on.
In each of the following sharpening steps, you'll raise the angle just a bit.
That way you're always shaving the stone with fresh metal.

Commentators to this howto are rightly pointing out the merits of a 20 degree knife edge, (knife held at a 10 degree angle to the sharpening stone) or a 17.5 degree wedge. I think my homemade plane blade sharpening fixture is set at 27.5 or something nerdy like that.
Use your own numbers, not mine, and by all means get carried away with your own refinements.
The numbers I picked aren't too important, just that you raise them with each step.

Step 6: The Sharpening Finally Begins

Flip the stone over and stroke the blade edge forward at a 6 degree angle. First one side of the knife, then the other. You are cutting toward the stone.

Step 7: Gilding the Lily

You've already endangered your friends by putting on an edge on a knife they're expecting to be dull.
Now it's going to get even sharper. Get a piece of 600 grit emery paper and put it on a piece of glass.
Any other really flat thing will do, but glass is most popular. I've seen Klingit and Mayan woodcarvers use this method.
Stroke forward at a 7 degree angle, alternating sides. A couple of strokes is plenty, because you're taking off a miniscule amount of metal. If you're silly or special you could get finer grits up to 1200 and repeat.

Hats off to the commentators for true facts about edge angles. Unless you're into artillery in a big way, most of us will overestimate small angles. Your 7 will be more like 12 in reality. The important thing is to look at the edge, test it, and raise the angle til you're shaving just a little abrasive off with each step. Thick blades will naturally sharpen at the higher angles they were intended for. You'll never really thin a thick blade out that much.

Some things like plane irons and chisels benefit from a straight bevel. For that make a jig to set the angle. Plane irons seem to inspire the greatest nerdiness in people.

Step 8: Cut Your Leg Off

Not.
This step is a totally unnecessary way to show off. You can impress people this way, especially if you rip your leg open, blood gushes everywhere, and they have to take you to the hospital.

Smear your leg with the abrasive paste you made by flattening your stone.
Then stroke the knife over this paste, sharp edge trailing.
This is called "Stropping". It takes the microscopic hairs off the edge to make it strong and extra sharp.
This is how you sharpen a razor, except sane people use a piece of leather called a "strop" rubbed with red garnet abrasive dust.

My Granddad used to beat his kids with his strop when they misbehaved. His son, my uncle "Bird Dog" tried to shave without proper instruction and cut a big gash in the strop. In the ensuing punishment the new sharp corner cut him, he started bleeding all over, and my Grandmother Nana came flying out of the house with strong new theories about corporal punishment. She was half my Granddad's size and twice as powerful. The kids were able to eat dinner sitting down for a while after that.

Step 9: Don Montague Tests the Edge

His style of fingernail testing involves resting the edge on his nail to see if it slides off or catches.

Step 10: Improvised Sharpening Stones

There you are in your friend's kitchen trying to cut a tomato with a blunt knife.
You're mashing it and smearing the skin around and not feeling sexy at all.
You ransack the utensil drawer and find not an abrasive.
You get the urge to criticize your pal for being the wrong kind of tool-using ape.
Halt.
Use an improvised stone instead. Here's a short list of what can work:

The underside of the toilet tank lid.
The rough unglazed ring on the bottom of a plate or other ceramic item.
an emery board used for manicures
A brick.
A flowerpot.
Any aluminum item. It's covered with a layer of aluminum oxide, a good abrasive. This method is only good for final sharpening.