Aristotle asserted that tragedy represented the highest form of art. In its capacity to help audience members purge unwanted emotions like fear and anger, it most fully accomplishes the cathartic aims of art in general and theater in particular. A good tragedy is able to maintain balance within society by providing an outlet for feelings (and therefore behaviors) which are pernicious or harmful.
The fact that tragedies incite feelings of pleasure in their viewers has perplexed critics and philosophers for centuries. How is it that, watching our deepest anguish or terror enacted on a stage (or, nowadways, on a screen), we can leave the theater feeling better than when we went in? Are we all a bunch of sadists, as some have suggested? Or is this the way we ought to feel?
Traditionally, a tragedy is any dramatic plot structure in which the hero's happiness is abruptly interrupted by a reversal of fortune brought on by some misdeed (either purposeful or accidental) on his part. This wrongdoing is almost exclusively the result of hubris—overweening pride. Pride, for the Greeks, was a two-edged sword; it was necessary for any man to acheive greatness, but it was also the universal source of his undoing.
Theseus' hubris led him to confront the Minotaur....and to abandon the girl who helped him get out of the Labyrinth alive. Oedipus' hubris helped him become king....and to kill his own father and marry his own mother. The hubris of Jason provided the impetus for finding the Golden Fleece....and the basis for his betrayal of Medea.
Some see the tragedies of Shakespeare as a radical departure from the Greek examples, that his device involved a hero's fall as the direct result of some sin he had committed. If Shakespeare hadn't stolen his plots directly from the Classical authors, this theory would be more plausible; it would also make more sense if hubris wasn't so closely linked to the Greek concept of evil. Sure, the heroes often came to do wrong unintentionally, but their hubris was never unintentional, and was always the ultimate source of their wrongdoing.
That's not to say Shakespeare didn't reshape the Classical plot structures to better reflect his own Christian-dominated cultural context. Whether ol' William was a Protestant or a closet Catholic is still undemonstrated (it's likely he was a hodge-podge, much like Queen Elizabeth during whose reign he wrote and acted), but he did grasp the worldview at work within the English society of his day. His tragedies, as a result, show evil punished and good (while not explicitly blessed) privileged.
Modern playwrites have largely abandoned the traditional tragic form. For most of them, either everything is a tragedy, or nothing is, and anything more ambiguous is seen as passe, old-fashioned. Which is ironic, given the current mood of relativism. Holding to Aristotle's rigid rules for a successful tragedy (outlined at length in his Poetics) is certainly unnecessary, but the new ultra-realistic plays rarely capture the cathartic element of tragedy and serve only as set-pieces for academic reflection and cultural snobbery.
A real tragedy gets at the heart of the human condition. It shows us, sometimes even rubs our faces in the lows to which each of us is capable of falling, and while it may offer a ray of hope, there's no magical resolved ending where everyone learns a lesson and becomes a better person. That's the audience's job—to learn from the mistakes of the characters on stage and try to avoid the same mistakes themselves.
Resolved endings aren't really cathartic, anyway; unless everyone dies, or the world ends, or Oedipus gouges out his own eyes, on what will we be able to project our own negative emotions and feelings? We need to witness horror and pain to get rid of our own. A good tragedy makes the audience work for any reward it receives, not just sit back and relax. The times we're able to emerge with a better sense of the world and ourselves are the times we can be sure we've witnessed a good tragedy.