The Apollonians
There are two types of Greek citizens in Nietzsche's calculation. The first is the Apollonian, the sober-minded man whose culture heralds the perfection of the physical form and, often, the ability to do combat.
The Dionysians
The cult of the Dionysians was a popular Greek religious group that celebrated festivals by the performance of high tragedies. This is the second class of Greeks, those concerned with the metaphysical prospect of transcending suffering, of making life glorious.
The Tragedians
By Dionysian tragedy, Nietzsche means the tragic plays written by such Greek Tragedians as Aeschylus, Sophocles, and the great Euripides. These artists are Nietzsche's pick for best artists of all time, because they seem to have a good grasp on the problem of life: Glory is required to make life worth living, even though it brings about judgment.
Aristotle
By discussing Greek tragedy, Nietzsche must address Aristotle, whose opinions of Greek Tragedy in particular helped shape the art of drama in the Western world. Aristotle's stance was that tragedy brings catharsis through sympathy. Nietzsche adds that tragedy is like a group experience and, therefore, it's an ecstatic experience.
