The cup is a unit of volume in the US customary unit system with the symbol cup.
It is equal to 1/2 US customary pint, 1/4 US customary quart and 1/16 US customary gallon. it is also equal to 236.5882365 millilitres.
Used primarily for cooking - the cup was adopted and established as a recognised unit of measure as it could be used by almost anyone in any kitchen. Due to the slight variants in design and size, it is rarely used as an accurate measure for important or critical materials, however it does represent a specific amount and can be used across many recipes to indicate what is needed.
The US gill is a unit of volume in the US customary unit system and uses the symbol gi.
It represents 1/4 of a pint - so in this instance, it is 1/4 of US pint - therefore measures exactly 118.29411825 ml. It's also equal to 8 tablespoons, 24 teaspoons and 5/6 imperial gills.
The gill has been mentioned a number of times in popular literature; including in Treasure Island, Animal Farm and in Dan Simmon's 'The Terror' - and all references are pointing to the use of the gill in terms of alcohol measures or servings.