In European tradition, knives are beveled on both sides of the blade; coming to a point in the middle. In Japanese tradition, however, the bevel is only formed on one side (usually the right-hand side). What is the purpose and/or advantage of this?
For vegetable knives, the main advantage of the front-side single bevel is that it's easier to make super-thin cuts. For example, a test of knife skill involves paring daikon radish to remove the peel, then continuing to cut to make a long continuous paper-thin strip, longer than you would be able to make if you just sliced it. See, for example, http://www.youtube.com/watch?v=oLZUJGyuhQM
People who have mastered this skill can also make remarkably fast matchstick cuts after making the long piece, and the lack of back bevel may reduce the amount of food that sticks to the knife. Some of these single-bevel knives that features a slightly concave backside (urasuki), which may also reduce the amount of adherence of food to the knife.
For crab knives, the honing is on the opposite side to the hand (the back of the knife), which may reduce the amount of shell fragments that could embed themselves into the meat, and probably allows the meat to cut more cleanly.
In practice, though, most Japanese households use dual-beveled knives. Single-bevel is more common in professional kitchens, especially in the Kansai area where fancy cutting techniques are used in presentation to a greater extent than in Tokyo or north Japan.
Although I have a nice nakiri knife and a cheap santoku, both are dual-beveled.
Knives meant for cutting regular fish or meat are usually, but not always, dual-beveled, the exception again mostly being for professionals. (I think yanagiba and sushi knives are usually single-bevel, but are again mostly used in professional kitchens and almost never at home). I thought sobakiri and udonkiri were typically single-bevel to help push the cut noodles away, but more recently I am less sure that the single-bevel is as common as I had previously thought for soba or udon knives.
The Japanese wikipedia entry for usuba knives suggests that the single-bevel edge is more durable, so, if true, that would be an advantage in high volume restaurant use. In my experience a thin, well sharpened Japanese blade may chip more easily than a thicker blade, so that resonates with me, even though I'm not 100% sure the notion that a single bevel is more durable is more than received wisdom.
With practice, I think you can get slightly more precision cuts with a single bevel than with dual-bevel knives, however, I don't think they are as easy to use for cooks with average knife skills.
Japanese vegetable cleavers are honed on one side only so the veggies don't stick to the knife and fall away as they cut then for stir fry, for example.
If you consider a western-style symmetrically sharpened knife, each side of the knife has a certain angle. For example, a typical mid-high end knife (ie globals) can be sharpened to 30 degrees inclusive, meaning each side has a 15 degree angle from the knife center. A smaller angle means a sharper blade and thus better at cutting.
If you took the same knife and sharpened it only on one side, one side would be flat (0 degrees) while the other would have a normal 15 degree angle. As a result the knife blade would have a 15 degree inclusive angle instead of 30 degrees for the same base steel, or in other words it's twice as sharp for the same price. Single-beveled knives also take less time to sharpen since the flat side only requires a small amount of work compared to a regular bevel. The disadvantages are that it takes practice to learn to cut straight with a single-bevel knife, and the knives are specifically left or right handed.
Historically this was probably a means of making the most of the poor quality steels that were all the Japanese could manage with their local resources. They knew that the quality of their steels was bad, which was also the origin of the peculiar folding and jacketing process used to produce katanas.
Note that many modern japanese knives are double beveled and asymmetric, which is more complicated to sharpen but may improve blade durability somewhat compared to a pure single-bevel.