The term "modern" can refer to the period in western civilization from the Renaissance onward, marked by the rise of the use of the vernacular or to the "modernist" period of the early twentieth century.
In the case of the more general use of the term "modern" as opposed to classical, The Playboy of the Western World is modern for several reasons, the first being its date. Next, it is written in English rather than Greek or Latin. It does not use a chorus and is written in prose. Female roles were intended to be played by female actors. It is designed for a proscenium stage with the illusion of a "fourth wall."
For modern in the sense of "modern drama" as it began with Ibsen and moved towards modernism, the play is broadly realistic, addressing the concerns of average people rather than nobles, kings, or heroes. Its plot is neither pure tragedy nor pure comedy but a mixture of genres. It deals with questions of sexual conduct in a way that would have been mildly scandalous in the Victorian era, and thus is "modern" in its treatment of gender issues.
The Playboy of the Western World is a modern play in the sense that it subverts traditional understandings of heroism. When Christy rocks up in the small rural town and proceeds to tell anyone who'll listen how he killed his father, the locals are filled with admiration rather than revulsion. All of a sudden, Christy is transformed into a mighty hero, earning love and respect in equal measure.
Ordinarily, parricide—the killing of a parent—isn't considered particularly heroic, and with good reason, too. However, in the context of a remote Irish town where nothing much ever happens, it's a different matter entirely. The locals are crying out for a hero to come and rescue them from their boring, humdrum lives. And Christy, with his lurid tale of how he dispatched his tyrant of a father, answers to their deepest needs. In doing so, he also turns the audience's notion of what a hero looks like completely upside-down.
J.M. Synge's The Playboy of the Western World is often considered a modern play because of it's representation of the rural Irish peasantry. Rather than depicting the rural Irish population as idealized and noble figures (as was popular at the time), Synge showed them as complex and flawed. Synge's peasants are at times bigoted and violent people (they idealize a man who "killed his father," after all), although they also show a real flair for colorful, poetic language and narration. As such, Synge doesn't paint rural Irish life in idealized tones, but instead tries to show it as a complex, flawed, multifaceted experience.
Though the play might seem tame now, Synge's depiction of Ireland's rural population caused an uproar in his day. The idealization of the Irish peasantry was an important component of the Irish nationalist movement, as Irish writers and patriots tried to use a heroic past and virtuous traditions to prove that the Irish had no need of English rule. While this tendency was noble in theory (it was, after all, part of an attempt to win freedom after long years of English rule), it also tended to blindly idealize the Irish peasantry in unrealistic ways. Synge's refusal to do so was highly rebellious, and indeed could be seen as a dangerously subversive attack on nationalist ideals. All in all, the urge to deconstruct the foundations of traditions is often a hallmark of modern literature, and so Synge's deconstruction of the idealized Irish peasantry is one of the main reasons Playboy is considered a modern play.