Chestnut is a common name for several species of trees in the genus Castanea, in the Beech family Fagaceae. Chestnuts are native to warm temperate regions of the Northern Hemisphere. The name also refers to the edible nuts produced by these trees. The nuts are an important food crop in southern Europe, southwestern and eastern Asia, and also in eastern North America before the chestnut blight. In southern Europe in the Middle Ages, whole forest-dwelling communities which had scarce access to wheat flour relied on chestnuts as their main source of carbohydrates.

The most famous depiction of chestnuts is probably their mention in the Mel Tormé and Bob Wells' classic, The Christmas Song which begins with the phrase "Chestnuts roasting on an open fire." Written in 1944, the song is most identified with Nat King Cole, although Tormé recorded his own version in 1965.

The Italian chestnuts are packed in 25 lb. cases - Italian origin generally has a higher fat content than the Korean or Chinese nuts - that's a good thing.

They are sized by count/pound (like shrimp)
AA -  the smaller chestnut, 60/65 count/ lb. 
AAA -  the larger chestnut, 48/50 count / lb. 

Other (US) growers measure them by size (diameter)

1.125" -- Small. These nuts are not sold to the public.Usually dry them and sell the product as either dried nuts or flour.
1.125-1.25" -- Medium
1.25-1.5" -- Large
>1.5" -- Jumbo
 
For FLAVOR - Michigan, California or Italian are the highest fat content and generally taste better - 
The Korean and Chinese cost less but are not known for their complex flavor profiles.