This is a great question. Some background information might be helpful here. The story of Antigone centers around her brothers and Creon, the ruler of Thebes. Polynices, her brother, wants to attack Thebes to take it over from his other brother Etocles. Both die in the ensuing battle and because Polynices attacks Thebes, Creon states that no one should touch the dead body of Polynices. The body should be left for the animals without the honor of a burial.
He also pronounces death for the one who would disobey. This is where the dilemma is. Should Antigone obey the command of Creon and allow her brother's body to be dishonored or should she disobey Creon and bury her brother at her own expense? Antigone decides to bury her brother. This fact shows us what she considers the worst crime to be - not honoring the dead with a proper burial, especially if it is your family. She would even be willing to risk death.