Finishing (also sometimes known as double matured) is the process of taking a whisky from one cask and then transferring into a second cask of different origin for a couple of months. There is no minimum or maximum time period for finishing, although typical finishing period is between six months and two years.
This 'secondary maturation' is undertaken to add complexity to the whisky. Typically, the first cask is an American oak barrel formerly used to mature bourbon, while the second cask may be one that has been used to mature some sort of fortified wine such as Sherry, though sometimes casks for Port, Madeira, or even Red or White wine casks or former Rum barrels are also used.
One of the reasons it makes such an impact isn’t just the changing of oak from American to European has on the spirit, 2-4% of the previous contents of the cask will remain in the wood after it’s emptied, allowing for the previous occupant to blend with the whisky that’s being added in as well.