Discuss the nature of tragedy in The Playboy of the Western World.
There may be a good reason you're having trouble figuring out the tragic nature of Synge's The Playboy of the Western World: it isn't a tragedy.
The play deals with conformity and rebellion, with myth making, and with mob mentality. It also presents a not-so-complimentary portrait of the Irish. But it isn't a tragedy.
Nobody dies, first of all. Second, fate doesn't lead to anyone's downfall. No life-changing battle is fought--figuratively or literally--by the protagonist. Nothing is cleansed away by a purge at the close of the work. In short, it's not a tragedy.
If one had to think in these terms, Playboy may be an anti-tragedy. The protagonist tries to kill his father, but fails; he is at first made into a hero for his rebellion, but later almost killed when the fickle crowd changes its mind; the hero is not a hero; the crowd's penchant for creating myths is exposed as faulty; and the play is more comic than tragic. It is almost mock-tragic, in the same sense as something mock-heroic or mock-epic.
The villagers are bound to tradition, prone to create heroes where there are none, and violent. Their early admiration for Christy's violent act is certainly a comment on the Irish mentality. But it isn't a tragedy. For that, you would need to look at Synge's "Riders to the Sea."