If I owe you a pound, I have a problem; but if I owe you a million, the problem is yours
A debt is something that you owe, and this can refer to a financial debt or to some other type of obligation instead. If someone saves your life then you are indebted to them and owe them your gratitude. If you have committed a crime then your debt is to society, and this is paid by completing the sentence that you were given by the courts.