The original definition of dramatic requirements for a Greek Tragedy were put forward by Aristotle in his Poetics:
Tragedy is a representation of a serious, complete action which has magnitude, in embellished speech, with each of its elements [used] separately in the [various] parts [of the play]; [represented] by people acting and not by narration; accomplishing by means of pity and terror the catharsis of such emotions.
You will notice that the "action" of the Tragedy, as described by Aristotle, must have "magnitude" and be performed in "embellished speech." Put simply embellished speech would refer to verse rather than the "everyday" language of prose. It is also meant to end in "catharsis," by way of "pity and terror" -- a culmination experienced both by the protagonist (tragic hero) and the audience.
For Aristotle, the tragic hero was meant to be noble, and he was referring literally to nobility of birth -- i.e. not a commoner. The hero (protagonist) should also be subject to a tragic flaw -- an aspect of his character that causes him to behave in a way that will lead to his downfall. For Aristotle, this must be a person who is "good," but makes an error, rather than a villain who is simply "bad" in his nature.
In modern Tragedy, some of these elements are different. We might even think of a modern Tragedy as a sort of watered down classical Tragedy. I'll use a commonly termed modern Tragedy, Death of a Salesman by Arthur Miller, to make my points.
This play is written in very vernacular prose, so there is none of the "embellished language" required by Aristotle. Willy Loman's downfall is caused by society, not by his own tragic flaw. He has become disposable in the world because of his age and the changing times rather than because of his own actions.
There is no catharsis of emotion at the end of this play. True, Willy dies and everyone feels sorry, but there is no terror, only pity. Feeling sorry for the tragic hero was a small part of what Aristotle felt a tragic ending should include. He insisted that there should also be a sort of epiphany of awareness about life as well. The understanding that Willy's time had come and gone was there from the beginning of the play, not a discovery made by audience and hero together at the end.
And, finally, of course, Willy is quite a "regular" guy, not some exalted or noble man.
Certainly, this is but one example of a modern take on the dramatic form of Tragedy. Please follow the links below for more information.