Irish Priest

Irish Catholic priests or nuns, basically, turn up in a lot of places. It's like Ireland exports them.

This was indeed once very common back in The Fifties and earlier when Ireland essentially did export clergy. In part this was due to the tradition of Irish priests and nuns going overseas both as students and missionaries and in part simply due to the very high emigration rate in Ireland (12.7 people per 1000 in 1950). It also runs with the stereotype, within Britain and the US, that Irish Catholics are particularly devout.

Having said that, this is definitely not the case today; changing demographics and the growth of secularism mean that these days, Ireland does not even train enough priests for her own needs, and has to import from Poland and Africa. Thus, unless the priest or nun is very elderly or is in a Period Piece, he or she is not too likely to be Irish. An exception is the United States, where even if priests who actually came from Ireland are fairly rare, Irish-descended clergy dominate the church hierarchy; there's a very good chance that any given American Catholic archbishop or Cardinal (both in fiction and reality) is Irish. However, the same is not true of the parishioners; religious surveys done within the last ten years show that Irish Americans are one of the most secularized ethnic communities in America. In addition, having been born and raised in America, Irish priests are unlikely to have an Irish accent despite what you find on TV.

Nevertheless, stock characters die hard, and even if doesn't make much sense any more, Irish priests abroad are not quite a Dead Horse Trope yet.

See also Christianity Is Catholic and Bad Habits.

In terms of rank, the Authority Tropes arguably equal are Badass Preacher, Corrupt Corporate Executive, Landlord, Preacher Man, Pedophile Priest, Schoolteachers Sexy Priest, Sinister Minister and The Vicar. For the next step down, see Student Council President. For the next step up, see Dean Bitterman.

Film

 * Father Brennan in The Omen.
 * Father Barry in On the Waterfront.
 * In Trading Places, Coleman dresses as one.
 * In The Delta Force, a heartwarming moment as Father O'Malley says that he should be grouped with the Jewish hostages, because he considers himself both a Jew and a Christian.
 * Father Everett in the Daredevil movie.
 * Father Geoghagan in The Wild Geese.
 * Father Alex in Mamma Mia!.
 * Father Janovich in Gran Torino. Despite the name, he's given an Irish lilt.
 * A very memorable one is played by Liam Neeson in Gangs of New York.
 * In Million Dollar Baby, the character played by Clint Eastwood keeps pestering an Irish priest with intentionally silly questions about Catholic doctrine, even causing him to drop an F-bomb.
 * Father Fitzgibbon in Going My Way played by Barry Fitzgerald defines this trope.
 * Father Peter Lonergan and Father Paul in The Quiet Man.

Literature

 * Mike Bennett's grandfather (he become a priest after he was widowed) Seamus in the James Patterson novel Step on a Crack.
 * Father Wolfe in A Tale Etched in Blood and Hard Black Pencil.
 * Father Luis Quinn in Ian McDonald's Brasyl.
 * Baby Kochamma falls in love with one in The God of Small Things.

Live Action TV

 * Caitlin O'Shaughnessy, in Airwolf, one pretended to be an Irish nun.
 * Nick impersonates one in an episode of Forever Knight. He gets to hear Schanke's confession.
 * Father Mulcahy of MASH is of Irish heritage.
 * Patrick Fitzpatrick of Veronica Mars. Also most likely a Corrupt Priest and a member of Irish Mafia.
 * Sons of Anarchy featured a prominent Belfast priest who was one of the top leaders of the Real IRA.
 * Ballykissangel Lampshades this since it's (initially) about an English priest assigned to a parish in Ireland.
 * One appeared in the Pedophile Priest episode of Law and Order SVU

Stand Up
"We used to grow priests in Ireland. We used to grow them from bits of people that we didn't like. But we over-planted. We had an epidemic. We were flooded with them. So, we tried to engage the rest of the world in a priest-for-potato swap. And we were conned by the Africans. Bastards! Took all our priests, not a potato between them. Pagan spudless fuckers! Our priests went over to Africa, and what happened? What do you think happened? They melted! And now we've run out of priests in Ireland. There's none left. And irony of ironies, what's happening? Missionaries! From Africa!"
 * Tommy Tiernan has a bit on this.

Video Games

 * Father Daniel Wales from BioShock (series) 2.

Western Animation

 * It seems that at least half of Springfield's Catholic clergy in The Simpsons is Irish.
 * And one of them is actually voiced by Liam Neeson.