Oedipus Rex the play is about more than just Oedipus and his seriously messed up family. It encompasses what happens after the myth, which is basically Oedipus' backstory. The play is considered a tragedy, and one of the main themes is man versus the gods. Also, the idea that everyone has an inescapable fate that is theirs alone plays a big part in the play.
Finally, in the Oedipus Inventory Oedipus Trilogy, there is Oedipus Rex, the myth. The myth is what adds most of the character to Oedipus Rex the character. The myth goes that Oedipus had a stake driven through his feet when he was just a baby, since he got a prophecy from the Oracle of Delphi that did not bode well at all for the kingdom. Then a servant who was watching brings Oedipus to the neighboring kingdom to be raised as a royal baby. The way they found him is the reason why his name is Oedipus, since it means "swollen foot". Oedipus then goes on the road, and on his way he kills his father in a fit of road rage, makes it to his old city, defeats the Sphinx by solving a riddle, and then wins his mother's hand in marriage as the prize for saving the city. That is about where the play picks up the story.
Next up is the Oedipus Inventory Aristotelian Trifecta (that sounds so cool). Aristotle himself was a huge deal in philosophy, and is still quoted and learned from today. He also was one of the first people to come up with a system of reasoning, otherwise known as logic. One of his greatest literary achievements was the idea of the tragedy, which is the idea that someone's life could just sort of generally degrade until there is not much good left and they claw their eyes out (only sometimes metaphorically) or that everybody dies except the one telling the story, which is rarely the main character.
Specifically, Aristotle's idea of a tragedy fits Oedipus Rex the play very well, as he was a great admirer of Sophocles's play, considering it the perfect tragedy. He said that a tragedy needs to have six parts; plot, characters, diction, thought, spectacle, and melody. Some of those seem more fitting to a song than a play, but to each his own. Aristotle's tragedy format is still in use, and is quite applicable to today's works as well as it was to the ones from BCE.
The final piece categorized under Aristotle is hubris. Aristotle mentions it in his description of a tragedy, saying that tragedy must include hubris amongst a host of other character flaws and problems. Hubris is a state, not entirely unlike arrogance, but also very different from that since it does not translate perfectly into English, but it is the opposite of being humble and striving to do your best. As soon as you believe you have reached arête (which is the opposite of hubris) then you have fallen into hubris, for you believe that your best is simply the best and that there is nothing you can't do.
Dramatic irony is when the audience knows something that the characters do not. In most cases, it is knowing how the play (or movie, or book, or whatever) is going to end, but sometimes it is something smaller like having an alternate perspective for a few moments so the audience knows what the bad guy is up to while the audience is oblivious. Perhaps the most famous story with dramatic irony is Romeo and Juliet, where the audience is literally told the plot of the entire play in the first paragraphs. There are many other examples throughout modern stories, also.
Sophocles is the guy who wrote Oedipus Rex. Without him, Language Arts classes everywhere would be so very different. But more importantly, he inspired a lot of people. Freaking Aristotle thought this dude was impressive for writing the perfect tragedy. He was one of classical Athens' three great tragic playwrights, and he was very attached to and involved in his city of Athens. Basically, he was a really famous guy of his time, and he was also really smart and really good at what he did.
Teiresias is important, and he also shows up in The Odyssey, another Greek classic. Teiresias was the blind prophet of Thebes. He saw Athena bathing, so she blinded him, but at his mother's pleas Athena granted him the gift of prophecy as compensation. He gave Oedipus his fateful prophecy, helped Odysseus with Poseidon, and made a prediction of the greatness of Hercules.
Fate plays a huge role in the story of Oedipus. The idea that there were three little-old-lady-celestial-beings-with-yarn making sure that you followed your destiny was a big belief with the Greeks. Everyone has their own prophecy, and nobody can escape their fate. The three fates determined what you were going to do, how much misery and fulfillment you got, and when you die. Cheery, right?
The final installment in the Oedipus Inventory is the Greek Theater. Greek Theater pioneered comedy and tragedy, now considered to be staples in any respectable theater. Plays were performed outdoors with a sloped seating area so everyone could see, and the seats curved around the stage so that the people could see every angle of the action. Also, no violence was ever shown on stage, just implied. Most of the great Greek plays are still considered classics today, and are still enjoyed by many.