Any competition to accomplish a task in the least amount of code. Usually counted in characters, but counted in lines if you're using a sufficiently gross language. Dense languages are favored; or, conversely, language density is sometimes defined by which language wins...
This includes PerlGolf, ForthGolf, and various multi-language games.
GolfScript was invented for CodeGolf and approaches the density of HQ9+ (a joke language designed for code golf...) while solving 'real' problems.
YouMightBeaThreeStarProgrammer if you golf at all, but especially if it affects your day job...
There are various sites that support code golf.
A promising new site is the StackExchange site at http://codegolf.stackexchange.com/questions/tagged/code-golf. It's an old-fashioned solution - the only "win" condition is upvotes from peers, and there's no automated testing of any kind. (Or maybe it's a cutting edge solution leveraging Web 2.0 and the social network instead of mere technical prowess...) The shortest entries there tend to be in JayLanguage, GolfScript, or occasionally HaskellLanguage or PerlLanguage.
SphereOnlineJudge supports code golf on a very small number of questions.