Beyond the "Ay" being used in an analogous way as when we use "thee" for "the" it is used in other ways:
It is used when it is the beginning of a title of a book, movie, play, poem or whatever: A Midsummer Night's Dream; A Christmas Carol, etc.
There are some I don't understand the pattern of, but they sound right: A dozen eggs (I have heard that it is when you are emphasizing the "one" meaning)
A once great nation (could be because of the grouping of people that makes a nation, but it might be for reverence)
A Mrs. Reynolds called last evening. (The speaker is relaying the message but does not know Mrs. Reynolds.)
I think it is more common in warnings or dangers when you are emphasizing the uniqueness of a situation or the adjective preceding the noun. And it is used as a mild expletive or rather a subtle suggestion that one might easily have been inserted: This is quite a predicament!; You are walking a very fine line buddy!; Watch yourself! There is a very big dog on Harold's farm.; We were on the path, when a frightening wombat lunged at Vivian.; Keep your eyes open! There is a crack shot sniper that favors that hill.