Correct spelling, explanation: this word’s spelling comes from the French word feide (hostility). Fued is the wrong form, but the misspelling probably comes from pronunciation, which seems more intuitive. Feud is, though, the correct form.

Definition of feud:
noun, long-lasting disagreement between some people
The Capulets and the Montagues were feuding which was the reason for Romeo’s and Juliet’s tragedy.

Incorrect spelling, explanation: although it might sound deceiving because pronunciation rather points to this spelling, it is incorrect, since the origins of the correct spelling are French. The original Old French word is feide and that is where feud, the right form, comes from.