Off to World Wide Words. And here I find the entry:
We are in Ireland, in what was once a village on the high road out of Dublin but which is now one of that city’s suburbs. King John gave a licence in 1204 to hold an annual fair there.
By the eighteenth century it had become a vast assembly, held on August 26 and the following 15 days each year, a gathering-place for horse dealers, fortune-tellers, beggars, wrestlers, dancers, fiddlers, and the sellers of every kind of food and drink. It was renowned in Ireland and beyond for its rowdiness and noise, and particularly for the whiskey-fuelled fighting that went on after dark. A passing reference in, of all sober works, Walter Bagehot’s The English Constitution of 1867, gives a flavour: “The only principle recognised ... was akin to that recommended to the traditionary Irishman on his visit to Donnybrook Fair, ‘Wherever you see a head, hit it’.” The usual weapon was a stick of oak or blackthorn that Irishmen often called a shillelagh (a word which derives from the town of that name in County Wicklow). The legend was that visitors to Donnybrook fair would rather fight than eat.
As Donnybrook progressively became a residential suburb of Dublin, the fair became more and more a nuisance until a campaign was got up to have it closed; in 1855 the rights to the fair were bought up by Dublin Corporation and it was suppressed. It was around that time that its name started to be used to describe a brawl, at first in the form like Donnybrook fair but then elliptically.
And here's a "fighting song" ;-)