Something that is off the cuff is unplanned or done on the spur of the moment. The phrase usually relates to impromptu speech, but it can also relate to anything else that is improvised or done on short notice. The phrase usually functions adverbially, in which case it does not need to be hyphenated. But when it’s an adjective preceding the noun it modifies (e.g., off-the-cuff remarks), it is hyphenated (according to the conventions for phrasal adjectives).
The first example listed in the Oxford English Dictionary is from 1938, and the earliest examples uncovered in historical Google searches are also from the 1930s. The phrase might derive to the practice of making notes on one’s cuff in last-minute preparation for a speech, but this origin isn’t definitively established, and there are a few other, less plausible theories.
Bruning is known for his off-the-cuff, occasionally controversial remarks. [Washington Post]
Johnson, in Clapham Junction, saw an off-the-cuff speech interrupted by cries of “Where were the police?” [Guardian]
Speaking off the cuff for 10 minutes in the middle of his set, he defended his most recent music video, “Monster”. [New York Times]
But if you’re not so lucky, you’ll be peppered with questions and need to keep your composure while forced to speak off the cuff. [Consumerist]