First, the word finish is a term used in whiskey production where it refers to the type of whiskey that is transferred from the primary maturation barrel or cask and put into a secondary barrel or cask. This transfer aims to make the whiskey undergo secondary aging with the main aim of attaining an intended finish. For instance, by definition, bourbon is matured in new oak barrels. After this maturation process, the resulting bourbon can then be further stored or finished in other types of used barrels. Those barrels could have contained wine, port, or rum, etc.
Secondly, the term finish also means the aftertaste you feel in your mouth when enjoying a particular type of whiskey.