A hone is a stone for sharpening blades. A blade is moved over the stone, which grinds away metal and in that way sharpens the edge. Hones come in a variety of sizes, from very small natural stones measuring less than an inch in width and just a few in length, to the large 3x8 inch (and sometimes even larger) stones.
Natural honesAs mentioned hones can be either natural quarried stones, or man made. Popular natural stones are the Belgian Blue and Coticule from Belgium as well as the Thuringer and Hunsrueck type stones from Germany. Besides these there are others, like the Dalmore, Tam O'Shanter and Water of Ayr hones from Scotland, and the various Arkansas hones from Arkansas (no surprises here), the Chinese finishing hones, and a wide array of hones from Japan.
Artificial honesFor reasons of cost, consistency and reliability of supply there are a number of man-made hones availble. Most artificial hones are made from some sort of ceramic base with abrasive powder mixed into it before hardening. Norton, Shapton and Spyderco all make this type of hones, barber hones are also of this type. Another type is the diamond plate like the ones DMT make, which is a metal plate with a thin layer of diamonds.
Grit sizesMost manufacturers/miners give out an (approximate) grit size for their hones, the higher the grit number the finer the hone. Unfortunately there are several different ways of measuring grit sizes, and even when adhering to the same standard accurately measuring grit sizes is difficult. Because of this grit sizes should be viewed as a rough guideline as to a hone's coarseness. Grit 1000 from one manufacturer may be much coarser than a grit 1000 from another.
In addition to this two hones of the same grit may cut (i.e. remove metal) at different speeds as well. The hardness, how fast or slow a hone wears down, also varies between hones.
Grit measuring standardsThere are three common grit measuring standards maintained by three organisations: CAMI/ANSI (American National Standards Institute), FEPA (European Federation of Abrasives Producers) and JIS (Japanese Standardization Organization). The three have different ways of measuring and different levels of strictness, leading to different ratings of the same stone. There are several sites dedicated to comparing different standards such as UAMA's page on grit standards, Dieter Schmid's conversion chart and the Society of American Silversmiths abrasive comparison chart.