Gaffe, with an e, refers to (1) a clumsy social error, (2) a faux pas, or (3) a blatant mistake or misjudgment. The far less common gaff, without the e, has several definitions related to fishing and sailing, and it also bears the sense a disreputable music hall or theater, which is used primarily in British English.
Examples
Older people have more trouble detecting social gaffes committed by others, the result of a decline in how they perceive emotions. [Reuters]
The Tribal Warrior is a 15.4-metre gaff-rigged ketch. [Sydney Morning Herald]
Nicolas Sarkozy, the French president, was jeered after an embarrassing gaffe in which he suggested the historically disputed Alsace region was still in Germany. [Telegraph]
The stepson said he remembered the fishing gaff hook hanging from his backside after his stepfather threw it at him … [The Sudbury Star]