Derived from the name of an annual Irish fair (until 1855) called Donnybrook Fair, in County Dublin, which was famous for its drunken dissipation and public brawls – the word is now used to refer to any particularly fractious free-for-fall or wild fight.
An imbroglio, on the other hand, derives from the Italian, imbrogliare, to entangle, and is used to describe a particularly complicated or violently convoluted mess of circumstances, usually embarrassing or scandalous. The Italian word, which has been adopted into English but is sometimes also expressed in the clumsy, Anglicized ’embroilment’, is from Middle French embrouiller, from Latin brodiculare – to broil.
A Brit equivalent for imbroglio is argy-bargy, but this is used more to express a lively, disputatious discussion rather than an imbroglio or, heaven forbid, donnybrook. Evidently derived from the Scots argle, meaning argument.
Argue, interestingly, and according to etymologists, derives partly from argutus, meaning ‘clear’ and argentum, meaning ‘silver’ – so the idea of argument, among the civilized ancients, was to bring greater clarity to a discussion. Not to precipitate a donnybrook.