Whetstones are a very unique item that allow a player to provide a weapon with a certain level of Sharpness, guaranteed, but dependent on how long the player wishes to gather Whetstone Powder. Whetstones can be made using three layers of materials; Sand of any variety on top, Petrified Wood (the item, not the planks) in the middle and Umberstone on the bottom.
To enchant (aka improve) a Whetstone, you must surround it with eight Whetstone Powders, acquired by killing Woodlice. This will increase the level of Sharpness that can be applied by a Whetstone. In order to use a Whetstone on a weapon, you must combine it with the weapon on an Anvil like you would any Enchanted Book. Of course, just like an Enchanted Book, the Whetstone will be destroyed by use, but will provide the weapon with the chosen level of Sharpness (which, for anyone using Sharpness, is only ever going to be Sharpness V).
Apart from this unique ability, Whetstones have no purpose.