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.