The noun debt refers to an obligation to pay for or do something. If you get arrested for stealing, serving time in jail is the way to repay your debt to society.
Debt comes from the Latin word debitum, which means "thing owed." Often, a debt is money that you must repay someone. Debt can also mean the state of owing something — if you borrow twenty dollars from your brother, you are in debt to him until you pay him back. If someone says, "I owe you a debt of gratitude," it means you have done something great and that person intends to do something to "repay" your kindness.