It has several different meanings. The most literal meaning is as an adjective to do with cork, and cork is the wood of the cork oak. It's because thie wood is used to make stoppers for wine botles that the stopper is called a cork (even when it's made of plastic.) If the wine in a bottle has reacted with the wood in the cork, creating an unpleasant flavour, you say the wine is 'corked' - you can also say it's 'corky' if it just tastes somewhat like this (so it may still be drinkable,.)

Corky is also the name of a plant. As well as these two meanings, it's a popular nickname especially in the USA, and is sometimes even used as a given name. In this case it seems to be based on a popular strip cartoon character, but where that name itself comes from is less sure, However, in the early 19th century the word corky was sometimes used in slang to mean a lively, fun person, so maybe that usage survived as a nickname.