Concerning J. M. Synge's The Playboy of the Western World, the "Western" part of the title refers to the West of Ireland. The play stems from a story or stories Synge heard while on the Aran Isles, west of the coast of Ireland. The play features the peasant population of such a place.
More importantly, you should know that "Playboy" does not carry the same denotation or connotations the word carries for us today.
The playboy is what Christy becomes. When entering the stage for the first time, he is much like everyone else in the play. But the play is partially about myth making, and Christy soon gains the status of myth. The story of his rebellion against his father and supposed killing of his father, gains importance every time he tells it, and reaches the point of myth. Christy gains self-confidence, which is only enhanced by the domination he displays at the games.
Seen from a distance, Christy becomes the playboy of the Western world, an almost mythological and, by the way, Christlike figure, who carries the hopes of the peasants on his shoulders, so to speak. "Playboy," I believe, refers to his daring deed and athletic prowess, rather than his "clubbing" and financial situation, etc., that the term suggests to us today. Christy is a hero.
Of course, seen up close, when Christy kills his father in "their own back yard," the bloody deed loses its romance and illusion, and Christy loses his status among the community, although the self-confidence he gains seems to remain.
The significance of the title lies within the text of the play and within the irony the line represents. First, Christy has experienced a meteoric rise in popularity among the inhabitants of his adopted village and has found within himself a previously unsuspected skill at sports games. In fact, he wins every local sports contest. It is because of this the villagers give him the title Playboy of the Western World: he is the boy best at all sports play. This is illustrated by the Widow's remarked to Mahon while she is trying to keep him away from Christy:
WIDOW QUIN — [with the shade of a smile.] — They're cheering a young lad, the champion Playboy of the Western World. [More cheering.]
The irony lies on one hand in the fact he was beaten at sports and beaten by sports players in his younger days. It lies, on the other hand, in the fact that the villagers idolize him--and allow him the opportunity to claim his elevated title--after he attempts to kill his father and thinks he has indeed killed his father.
Synge is illustrating the irony and foibles of human consciousness that embraces behavior, ideas and people based on illusion--then let them go just as suddenly when illusion meets reality:
PEGEEN — [glaring at Christy.] — And it's lies you told, letting on you had him slitted, and you nothing at all.
The significance of "western world" in the title of The Playboy of the Western World by J. M. Synge depends on the specific meanings of the terms in Ireland at the beginning of the twentieth century. To the original audience, the "western world" would have meant western Ireland, an area including Galway and the Aran islands. These areas were considered a culturally iconic centre of Irish identity, where people still spoke Gaelic, were in touch with their Irish roots, and were least corrupted by contact with the Protestant Ascendancy. Thus a play which showed a certain critical approach to the "western world" would be like a play in the modern United States criticizing "Midwestern values" playing to an audience of conservative U.S. citizens who consider the Midwest the "heartland" of their country.