What happens to debt that you owe to a dead person?
It depends on the nature of the debt. If it is a formal debt, such as a contract for deed or a mortgage, the estate of the deceased will likely continue to collect on the debt. You will likely be contacted by the executor of the deceased estate to discuss payment of the debt.
You still owe the debt to that individual's estate. The obligation to repay a debt does not die with the creditor; if it did the debt would be a gift to the borrower, and that is not the case. The assets of an estate, including the debt you owe, will go first to pay the debts of the estate and then to the heirs, either under a will or under the state law on property inherited without a will (intestate succession).
There is a process for collecting the debt from you. If there was a will, the personal representative (Executor/Executrix) of the estate will be responsible for gathering the assets of the estate, including collecting debts. If there was no will, a survivor, usually one who would inherit in the absence of a will, would have to go to court and seek to be appointed administrator of the deceased's estate, make a demand on you, and, if necessary, sue you in the name of the estate to collect the debt. It’s possible that the deceased owed more money than there are assets, and the descendants will have no interest in handing the estate.
It might also be possible for you to settle the debt with the heirs by compromise. If you knew, for example, there were only two surviving children, no claims on the estate, and the debt is the only asset of the estate, you could agree to pay the two children directly a portion of the amount owed. If you paid them directly they could avoid the costs and time of probate. You'd explain there are risks to you in paying them, should other creditors turn up, that because of that risk you would expect a discount. If you pay them directly, you should still insist they agree to indemnify you up to the amount of money you pay them should there be other claims against the estate that you are currently unaware of (such as from the deceased's creditors, other heirs that may become known in a will you all are unaware of, or tax authorities).