The angels' share (also 'angel's share') refers to the volume of whiskey or wine lost to evaporation during the aging process. This evaporation takes place during the time that the beverage ages in wooden barrels, and is not an issue after it has been bottled.
There is some conflation between the evaporation of alcohol and of water in calculating the angels' share. Some sources refer specifically to the reduction in proof, others reduction in volume, and some flip indifferently between the two. However, estimates range from a 2-10 percent yearly evaporation rate for the alcohol (decreasing over time, and depending on the local climate), and up to 30% volume reduction between distillation and bottling. Barrels stored in low-humidity environments will lose comparatively more water-to-alcohol, resulting in higher proof; the converse is also true.
Many concoctions are very high proof and comparatively low flavor immediately after fermenting; the period during which the spirits sit and mellow is an important part of their production, and as such, losing some alcohol is not undesirable. Whiskey that is sold as high-quality 80- to 100-proof may start its life as 150-proof paint cleaner until it is properly aged.
While this term appears to be used all over the world, it is particularly associated with British Isles. This may in part be due to the 2012 movie The Angels' Share, which was set in Scotland.