Vermouth's most famous moment is as the lesser part of a martini (esp. gin martini). The portion of vermouth defines how wet the martini is, and most people prefer dryer martinis.
It has been noted that an ideal gin martini is a glass of gin drunk whilst looking at a picture of vermouth.
Along similar lines, during one of the early atomic bomb tests, a bottle of vermouth was strapped to the device so that in the future, all one need do to make a perfect dry martini would be to hoist a glass of gin to the sky. Bear in mind that the authenticity of this anecdote is in doubt, as I can't remember where I read it.
Vermouth's second most famous use is simply as a cooking wine.