Augureys appear in Fantastic Beasts and Where to Find Them; their entry reads:
Augurey (also known as Irish Phoenix)
[Ministry of Magic] Classification: [Harmless, may be domesticated]
The Augurey is a native of Britain and Irleand, though sometimes found elsewhere in Northern Europe. A thin and mournful-looking bird, somewhat like a small and underfed vulture in appearance, the Augurey is greenish black. It is intensely shy, nests in bramble and thorn, eats large insects and fairies, flies only in heavy rain and otherwise remains hidden in its tear-shaped nest.
The Augurey has a distinctive low and throbbing cry, which was once believed to foretell death. Wizards avoided Augurey nests for fear of hearing that heart-rending sound, and more than one wizard is believed to have suffered a heart attack on passing a thicket and hearing an unseen Augurey wail. Patient research eventually revealed, however, that the Augurey merely sings at the approach of rain. The Augurey has since enjoyed a vogue as a home weather forecaster, though many find its almost continual moaning during the winter months difficult to bear. Augurey feathers are useless as quills because they repel ink.
Fantastic Beasts and Where to Find Them
There's also a footnote remarking that noted loon Uric the Oddball kept about 50 pet Augureys, and subsequently went a little... odd.
This appears to be the only other information about Augureys in canon, although as ibid extensively cites, they were referenced a handful of times in Quidditch World Cup material on the old Pottermore; an oversized Augurey named Hans was the mascot of the Liechtenstein national Quidditch team.
As far as I can tell, Rowling has never remarked on her inspiration for the Augurey, though the description above calls to mind several possible (partial) influences:
The Phoenix, though not native to Irish mythology, is sometimes used as a symbol for the Provisional IRA
Dark birds, of course, have been death omens throughout history; there's a lot of superstition around crows for that reason
Similarly vultures, as with most scavenger birds, have a long cultural association with death and decay
The idea of the "mournful cry" is also a popular death omen, though in Irish mythology it sometimes takes the specific form of the banshee, a female spirit believed to foretell deaths in the Irish royal families:
Her mission is to give warning, by a plaintive wailing cry, of the near approaching death of one of the family, and this cry is repeated at the moment when the soul leaves the body. [...] The spirit is generally heard at night, sometimes at midday, and very rarely in the mourning. The mournful cry is generally the only indication of her presence, but in a few instances she has been seen for a moment as a rapidly receding figure having the appearance of a withered old woman clad in flowing white drapery.
The Funeral Customs of Ireland by James Mooney (1888), published in the Proceedings of the American Philosophical Society vol. 25
Rowling, of course, actually included banshees in her story, though they bear little resemblance to the banshees of myth; Rowling's banshee appears to be a mashup of the mythological creature with the pop-culture image of the shrieking woman.
As Neeshka notes in a comment on the question, "augury" is a form of divination, originating in ancient Rome, that interprets omens by observing the flight of birds.