The barleycorn grain motive is a popular decoration in 1st half of 19th century in United Kingdom. Chess sets with that decoration were called "Barleycorn".
The essence, or origin, for the name of these ornamental sets is the decoration with the grain. Sometimes combined with other motives. The type has been made from ca. 1820 onwards. The end is less clear: certainly till a bit after mid 19th century (the F.H.Ayres set is made after 1864) and possibly till end of 19th. It was thought that the technique for the decoration was introduced by the so called Holtzapffel lathe, but that can only create some specific numbers of corn repetitions and we count non-matching numbers. A discovery by restorer Alan Dewey.
The decoration is only applied to King and Queen. Rook is rarely decorated, and then usually with a different motive. Bishop has a deep slit and the Knight is round. Bases are round and "cogged" usually. The fanciest bases are called Van Dyke bases. Most sets have "reeded" balls on Queen, Rook and pawn. The pattern applied to Bishops as well.
"Barleycorn" sets are mainly made of bone, but ivory sets do exist as well. Rooks in ivory sets may have a stem. English wooden "Barleycorn" sets may not exist.
Very often sets are wrongly called "Barleycorn": 1. Ornamental sets with other motives, like leaves, an acanthus motive, or the very rare "Ropetwist" sets. 2. Undecorated sets where King and Queen have a fat barrel (mostly ribbed), the Rook is fat without stem but with finial and often a flag, the Bishop has a deep slit and the Knight is round. 3. Undecorated slim sets. 4. German "Nuremberg" sets of similar overall shape, but their Bishop with a small or absent slit, a simple slim Knight and bases much smaller as their English counterparts. 5. German decorated types (sometimes with faces). 6. German (or some say Dutch) decorated type where King and Queen lack a stem. 7. German type with open worked barrels of King and Queen. Although the latter 2 are not very often seen as "Barleycorn".