Finishing is the procedure that some whiskeys undergo whereby the spirit is matured in a cask of a particular origin and then spends time in a cask of different origin (generally 6 months to 2 years.)
Typically, the first cask is an American oak cask formerly used to mature bourbon. The second cask may be one that has been used to mature some sort of fortified wine.
In the United States, finished whiskeys are legally classified as a Distilled Spirit Specialty, although that term doesn’t appear on whiskey bottle labels. The formal definition is “any class and/or type of distilled spirits that contain or are treated with flavoring and/or coloring materials and/or nonstandard blending or treating materials or processes”, in this case the flavoring simply being wine left in the wood of the second cask – TTB (U.S. Alcohol and Tobacco Tax and Trade Bureau)
There is a debate in some sectors of the whiskey world of whether a finished whiskey counts as a true/straight whiskey, or whether it is a flavored product. However, if one were to consider it a flavored product, then all Scotch would have to re-defined as flavored whiskey, because all Scotch is aged in already used in a barrel that has already held bourbon. (Bourbon, in contrast, may only be aged in new oak.)