The Free Dictionary suggests that
an emphatic exclamation of approval
which indicates that the letter A can stand for agree, assent, approve, acquiesce, absolutely, or anything that expresses approval.
And Wiktionary suggests that
an expression of triumph, or joy. Usually in response to a piece of unexpected positive news.
which means the letter A can also stand for awesome, amazing, astounding, astonishing, awe-inspiring, appreciate, etc.
If the expression "F**king A" is used as a noun to describe a person,
He is a f**king A.
the letter A is probably an abbreviation of the term a**hole (instead of ass) as it is more commonly used with the adjective f**king.
And if the expression "F**king A" is used as an exclamation to express anger or surprise,
Aw, f**king A!
it can refer to "f**king a**hole" as mentioned above,
Aw, f**king A! (He/she ruined my plan!)
or it can simply mean that the person is too lazy to finish his sentence.
Aw, f**king a ...! (it can be f**king a dog, f**king a balloon or f**king anything you want!)
F**king A is also a play written by American playwright Suzan-Lori Parks.
F**king A is inspired by the novel The Scarlet Letter, written by
Nathaniel Hawthorne in 1850. F**king A shares several similarities
with this work; both of the main characters are named Hester and are
strong females with an unwavering love for their illegitimate child.
Both Hesters also bear the letter “A” as a symbol of how society
defines them: the modern Hester is an abortionist, the original is
branded for adultery.
The idea for the play came to Parks while she was canoeing with a
friend, when Parks yelled out, "I'm going to write a play, a riff on
The Scarlet Letter, and I'm going to call it F**king A. Ha, ha,
ha!" While Parks was initially joking, she couldn’t get the idea
out of her mind. She began working on the play, but after 4 years of
drafts and rewrites she still hadn’t come up with a workable plot. She
threw out everything except the title and the name Hester, and
considered throwing out the latter as well, but the voice of Hester
filled her mind and persuaded her to retain it and write "her" story.
So when a person mentions F**king A, he/she may be talking about this play.
John: Do you know what I watched in the theater yesterday?
Sam: F**king A?
John: Yeah, F**king A.