Some terminology...

• wet, moistened, humid, etc. "My towel is wet."
• whet, sharpen, amplify, "Whet your appetite."

Tools

• a work surface like a towel or a sink bridge
• sharpening or whetting stones, both coarse to begin and fine to end
• leveling plate, a very hard stone or metal plat to dress your stone

Stone grit

The grit of stone to buy depends on the hardness of the steel in the knife being sharpened. As a general rule, good-to-excellent quality European steel tends to fall between 52/54 up to 58 on the Rockwell Hardness Scale while Japanese steel falls between 58 and 66/68 and will scratch glass (which is typically 60 on the scale).

• 52/54 → 58 use stones of 2,000 to 4,000 grit
• 58 → 66/68 use stones of 6,000 to 8,000 grit

For what most people have, unless they're sushi chefs with very exclusive knives, two stones 4,000 and 6,000 are perfect.

Steps

1. Soak whetting stone until stops bubbling.
2. Position coarse stone on a stable surface (like a damp towel or, better yet, a sink bridge which lets you do the whole thing over your kitchen sink), wet it, use the leveling plate to re-level the stone and work up some "mud" with which to sharpen.
3. Hold knife in dominant hand with the blade facing back toward you. The face of the blade should be at a 15-20° angle relative to the surface of the stone, and the length of the blade should be at a 45° angle relative to the length of the stone. Put fingers of other hand on the face of the blade and use them to push the edge down into the stone.
4. Work the blade back and forth at an angle that feels like shaving little bits of sand off the stone surface. When pushing the knife back and forth, slowly draw it perpendicularly across the stone to make sure the whole length of the edge gets sharpened. Re-wet the stone as needed to keep the edge moving smoothly and wipe the blade clean occasionally.
5. When a consistent burr across the the whole edge can be felt, it's time to sharpen the opposite side. Rotate the handle of the knife hand so the edge is now facing away, and this time hold the length of the blade at a 90° angle perpendicular to the length of the stone. Do just as for the first side.
6. Once a burr can be felt on the second side, return to the first side and repeat the process a few times with progressively fewer passes and lighter pressure. Be sure to work both sides of the knife equally, to keep edge geometry symmetrical.
7. Switch to fine stone for finishing. Wet it and level it with leveling plate.
8. Perform the basic motions all over again on the finer stone, but use less pressure. Done when no burr can be felt.
9. Strop both sides of the blade by doing the same motion one way only, moving the knife away from the direction of the edge.
10. Clean the knife and test it by cutting a hanging sheet of paper. If the cuts are clean and there are no tears, that means the whole length of the blade is well sharpened. If there are tears, that's an indication of dull spots or chips.