In Joyce's short story, "The Dead," the Three Graces are Gabriel's aunts and cousin and are likened to the three graces from Greek mythology. The setting of the story is Dublin at the Misses Morkan's annual dance. Gabriel is chosen is give a speech at the dinner. At the onset of his speech, he names Aunt Julia, Aunt Kate, and Mary Jane the "Three Graces of the Dublin musical world."
This name appears apt, for Aunt Julia is a soprano who is now near death, Aunt Kate taught and played piano, and Mary Jane is a talented pianist. All three women are musically inclined and have spent their lives working in the musical profession. Gabriel offers a toast to the Three Graces celebrating their lives and work:
"Let us drink to their health, wealth, long life happiness and prosperity and may they long continue to hold the proud and self won position which they hold in their profession..."
However, it is later in the story that Gabriel ponders Aunt Julia's death when he comes to the realization of the brevity of life. The Three Graces could be one reason that Gabriel ponders his own mortality and his own identity.
To understand Joyce's allusion, you should know a bit about the classical graces to whom he is referring. The Three Graces, celebrated in classical literature and art, were the daughters of Jupiter (or Zeus in Greek mythology), and companions to the Muses. Thalia (youth and beauty) is accompanied by Euphrosyne (mirth), and Aglaia (elegance).
We think of the Muses as young and beautiful but it is clear that these women have gone past their prime. The overall impression is that they are "fussy." Aunt Julia is described as a "woman who does not know where she's going." Kate is "too feeble to go out" and Mary Jane, once a music teacher, has long been surpassed by her pupils. The women appear to live comfortably, but their fussing seems rather pointless.