"Tragedy" means "goat-song," from Tragos "goat" + oide "song."
One theory about why tragedy is "goat-song" is that the first "tragedies" were sung by men in goatskins. In this theory, tragedy arose from Satyr-dithyrambs. Satyrs are half-man, half-goat. A dithyramb is a form of poetic song. Arion, the singer mentioned in Herodotus who was saved by dolphins, is reputed to have invented the Satyr-dithyramb, which had mime elements.
Another theory, that of Walter Burkert, is that tragedies were originally performances about or accompanying sacrificial rites, in which goat-sacrifice was prominent.
Tragedy was performed in honor of Dionysus, the god of wine. The content of tragedy, however, does not always have to do with Dionysus (it seems that at some stage the chorus' members ceased to be satyrs).
Essentially, the origins of tragedy are murky and it is probably wrong to insist on one point of origin. The place for tragedies was Athens, but the ancient sources point to Sikyon as the place of origin (on Peloponnese near Corinth).
The Dithyramb was a song sung by a chorus in the cult of Dionysus. About 531-535, a man named Thespis took the first step towards creating what we know as drama. He added spoken parts to the song, which involved adding the first actor (called hypocrites in Greek).
Aeschylus added the second actor: there are a few scenes in the Persians, the first surviving tragedy, which require two actors (417-433: Queen and Messenger; 1090-1383: Queen and Ghost of Darius). Sophocles added the third actor. Aeschylus also increased the role of the plot in tragedy and lessened that of the chorus. All the actors were male, and one actor could play more than one part.
The basic structure of a tragedy, at least in Aristotle's (Poetics 1452b14-27) account, is as follows. The chorus' entrance is called the "Parodos." Choral songs in the middle portion are called "Stasima" (singular "Stasimon"). You should imagine these parts as accompanied by music. The chorus danced during these parts as well. The actors' parts are the "Prologos," which we can call "the first act"; "Episodoi" (singular "Episodos"), which we can call "acts"; and the "Exodos," which includes any action after the last stasimon. The spectacular songs were the chorus' parts, which means that the chorus in ancient tragedy was probably not ponderous and awkward as it seems to be in modern productions. For ancients, it was central. In general, think of ancient tragedy as much more like Wagner than Shakespeare. And yet, dance was so important that Wagnerian opera is not appropriate either: perhaps something like Riverdance crossed with Wagner and Shakespeare. As time went on, particularly with Euripides, aria-like songs were used more. Particularly items that were typical in tragedies were messenger speeches (to announce what was not acted out on stage) and speech contests between actors. Also, stichomythia is found frequently, where each actor would deliver a single line followed by a single line of the other actor. Recognition-scenes were also frequent, wherein the (hidden) identity of an actor would be discovered by another actor.
Tragedy in Athens was not just a performance. It was also a contest (perhaps starting in 535, but certainly since 508), as well as a religious and civic event. The contest took place at the Great Dionysia Festival (and later also at the Lenaia). The state sponsored the festival, and rich Athenians under wrote the plays (called "choregoi"). The first day was given over to dithyrambs, the second to comedies, and the third, fourth, and fifth day to tragedies. The eponymous Archon chose three tragedians per year. Each playwright took one entire day for his plays. Tragedies were performed in groups of three tragedies followed by a satyr-play. The tragedies were more serious, while the satyr play was lighter fare. We have only one surviving satyr-play, the Cyclops of Euripides. Three tragedians were selected to write plays each year, and so there were a total of 9 tragedies and three satyr-plays per year. If you add it up over the fifth century, an astounding number of plays were written and performed in Athens (we have only a small fraction of them complete and a somewhat larger fraction of them in very fragmentary form). After his death, Aeschylus' plays were allowed to be reperformed, but until 386, when a special occasion for reperformance of old tragedies was created, no other tragedies were reperformed at Athens. The tyrant Hieron of Syracuse, however, asked for a reperformance of the Persians while Aeschylus was alive. At each festival, a jury would select the best playwright, the best actor (instituted in 449), and the best sponsor (choregos).
The actors all wore masks, wigs, and the cothurnos, a sort of boot-sandal with straps. Evidently the costumes for the Persians were impressive. Aeschylus had a reputation for impressive costumes. The masks will have been simple, not the grotesqueries of later times or comedy.
There were three sorts of line delivery. 1) Unaccompanied verse (iambic or trochaic), 2) recitative with instrumental accompaniment (somewhere between spoken verse and song: anapestic meter), and 3) song with instrumental accompaniment and dancing (in a variety of lyric meters).
Ancient theatres were semi-circular. Seats of honor were first row center. There was no stage per so. There was a building called a "skene" (from which we get "scene"), which had originally been a tent. It was the dressing room. There were three doors on its front. In front of the skene there was an orchestra (it's the source of our word, but it means "dancing ground"). It was circular. There the chorus and actors performed. In some plays there was a device like a crane that could deliver an actor/dummy from the air, which we call "deus ex machina." Euripides used it a lot at the end of his plays. The Theatre of Dionysus at Athens held 17,000.
The three great Athenian tragedians were Aeschylus (525-455BCE), Sophocles (497-405BCE) and Euripides (480-406BCE). Aeschylus was a soldier at Marathon and Salamis. He wrote about 90 plays. He introduced the second actor and reduced the role of the chorus, thus making "drama" possible ("drama" is from Greek dra- "do," and so "drama" is "doings," or "plot." We have seven of his plays surviving: Persians, Seven against Thebes, The Suppliants, Prometheus Bound, Agamemnon, Choephoroi (Libation Bearers), and Eumenides. Sophocles wrote about 125 plays. He won first prize more often than the others. He added a third actor, reduced the choral role further, and added scene painting. The extant plays of Sophocles are: Oedipus Rex, Philoctetes, Women of Trachis, Electra, Oedipus at Colonus, Antigone, and Ajax. Euripides wrote about 90 plays. He was the most interested in philosophy and intellectual currents of his day. He did not win as many prizes as others. We have 18 or 19 of his plays.
The reason why we have the plays that we do is that they were selected by schoolteachers of later ages as those that were appropriate for students to read and they were anthologized.
The Persians, produced in 472, won first prize. It is the first extant play of Aeschylus and the earliest extant tragedy we have. The wealthy sponsor of the Persians, its choregos, was Pericles (this is the first time Pericles appears in history). Aeschylus had been writing tragedies since 499. It divides into four parts:
The Persian War quickly assumed mythic proportions in Athenian minds. It is depicted in all art forms, much like mythology. Other historical events did not receive such treatment. Hence it is probably wrong to speak of "historical tragedy": rather, the incredible events of the Persian Wars became like myth. So we might speak of the tragedification of history, to coin a monstrous word.
"Responsibility" is an important theme in Greek tragedy. The gods apparently cause human affairs to be the way they are. An oracle foretells that Xerxes will fail (line 739f.). But that does not reduce the responsibility of humans for their own actions. By crossing the prescribed limitations of humans, a human commits an act of hybris, which is caused by ate (psychological blindness, failure of judgement). Sometimes excessive wealth and luxury lead to hybris as well. Hybris is an act of violence that violates prescribed norms. Xerxes' effort to conquer Greece seem like paradigm instances of ate and hybris, especially given the engineering projects to bridge the hellespont.