It is a bit confusing. There is a cask conditioned beer and there is a cask finished beer.
Cask conditioned beer is one that is carbonated without additional CO2. It's put into a closed container, generally a stainless steel cask, and conditioned by the CO2 formed by the residual fermentation or additional 'sugar' is added to specifically add some carbonation to the beer. It's then served straight from the cask. Cask ale has a lower level of carbonation than forced carbonated beer. It's similar in process to bottle conditioning, but a different result. The beer is smooth and very flavourful. Very popular in the UK and increasingly popular in North America.
The issue with trying it is that it's served fresh and spoils after a few day of being tapped, since air is introduced into the cask rather than CO2 from a tank. So ... you'd have to go to a pub to get it; not many places in NA serve cask ale. The interwebs could tell you whether there are pubs that serve cask ale where you live.
The beer that greenbasterd is talking about is beer that is aged in an old whiskey or other type of liquor cask. It adds a character from the liquor that was in it or the oak of the cask. Innis and Gunn is a good choice.
