This play takes place after the events of Oedipus the King. After Oedipus leaves Thebes, his sons (Eteocles and Polyneices) vie for control of the throne. Polyneices is ejected from Thebes through some trick. He then attacks the city and is killed in the battle, as is Eteocles.
Creon, the regent, makes a decree that Eteocles, the cities defender, will have all the normal burial rights for is status, while Polyneices, the attacker, should have none.
Antigone, the sister of the two brothers, refuses to follow this edict and insists on giving Polyneices the full burial rights. Again, as in the story of Ajax, the importance of burial, afterlife, and legacy don't fully translate to the modern sensibilities. Antigone goes ahead with this and is caught. Creon sentences her to death, and sends her off to some dungeon.
Creon's son Haemon is also Antigone's husband and he begs his father to relent. There is also a prophecy warning Creon to relent. Eventually he does, but too late. Antigone has already hung herself in the cell. Haemon kills himself upon learning of his wife's fate. Finally, Creon's wife, Eurydice, kills herself due to losing her son.
Creon ends the play a broken man.
There is a point here about pride and stubbornness and perhaps too the divine law (of burial) as superseding human law. Fate, family loyalty, tragedy are all here. It's a brutal play. Antigone is an epic hero, facing death for her principles. There is also this Romeo and Juliet aspect of Haemon killing himself over his love that was so close to being saved. These stories are as old as time.