ABV is an abbreviation of Alcohol by Volume - the standard measure of how much alcohol is contained in a given volume of an alcoholic beverage. It’s always expressed as a volume percent.
It is calculated by the number of millilitres of pure ethanol present in 100 mL of solution when at 20 °C. It is impossible to calculate the ABV without knowing both the density of the liquid and its temperature. All distilleries face severe fines if they calculate it incorrectly, with tax authorities concerned about anything being bottled over the stated amount, and trading standards when bottled below, with only a small margin or error is permitted (0.2% in the UK)
The term “Proof” refers to the same idea – how strong the spirit is – but is largely reserved to the USA (the European Union prescribed a standardisation in 1980 and so anything sold or made here must have the strength stated in ABV). In the United States, alcohol ‘proof’ is defined as twice the percentage of ABV.
Many spirit categories will have a minimum and maximum ABV they are allowed to be bottled at, and many will also define requited parameters in the production process too.