It is quite common for people living at the present-time to call every wooden cask by the name: barrel. But not all wooden casks were actually barrels. There were at least seven sizes of wooden casks and each one had its own distinct name.
Before noting the casks' names, we need to determine what we are talking about. In the days before plastic, vinyl and glass containers, both dry and wet goods were contained in either cloth sacks or wooden casks. Dry goods and materials could be carried and stored in either cloth sacks or wooden casks. The cooper who made wooden casks and other containers to hold primarily dry materials was called, appropriately, a dry-cooper. Wooden casks produced for the storage of primarily wet materials was made by either an ordinary cooper or a 'wet-cooper'. The difference being that the wet-cooper made items with both 'heads' intact while an ordinary cooper also made pails and buckets and items such as ladles. Then there was the 'white-cooper' who made items primarily intended for dairy products.
The various sizes of wooden casks that were common during the period of the American Revolutionary War were the 4-1/2 gallon pin; the 9 gallon firkin; the 18 gallon kilderkin; the 36 gallon barrel; the 54 gallon hogshead; the 72 gallon puncheon; and the 108 gallon butt. Then, as now, the 54 gallon hogshead was the most common for most purposes. There was another, less used cask known as the tun, which would hold up to 256 gallons. All of the other casks were divisions of that quantity of wet gallons.
The etymology of the word 'pin' is not so easy to determine. In the Latin, a pinna referred to the scoop of a water wheel. Perhaps the amount of water a mill's water wheel would scoop up amounted to 4-1/2 gallons?
The word 'firkin' derives from the Middle Dutch term vierdekijn which meant literally a fourth, or a fourth part. It would correspond to a fourth of a 'barrel' or nine gallons.
The word 'kilderkin' derives from Middle English, which derived from the Middle Dutch term kindekijn which meant a half and referred to a half of a barrel, or eighteen gallons. The Middle Dutch word kindekijn was itself derived from the Medieval Latin quintale which referred to one hundred.
The word 'barrel' derives from the Old French term baril which referred to a cask or vat. The Old French might be derived from the Proto-Germanic term barilaz referring to a jug or container, which itself was derived from the Proto-Indo-European term bher, meaning to transport or carry.
The word 'hogshead' was derived from either the Dutch oxshoofd, the Danish oxehoved; the German oxhoft or the Swedish oxhufvud, all of which referred to a large cask.
The word 'puncheon' derived from the Middle English term ponson, which was derived from the Anglo-French term ponchon which referred to a wine vessel.
The word 'butt' derived from a Middle English term that itself derived from the Anglo-French term but or bout, derived from the Old Occitan (Old Provencal) term bota, from the Late Latin term buttis which referred to a cask or barrel.
Something that is difficult for most people to grasp is the concept that despite the measurement of gallons for a wooden cask, both dry and wet materials held within a wooden cask would be weighed in pounds for sale or transport. For example, a hogshead of beer, being roughly fifty-four gallons, would weigh about 450 pounds since one gallon of water weighs approximately 8.33 pounds [54 gallons (x) 8.33 lb/gal (=) 449.82lb].
These are all approximate measures, of course and should be considered as estimates. A quantity of Ale would not weigh exactly as much as the same quantity of Beer, and neither would weigh exactly as much as the same quantity of Whiskey or Rum.