If a person dies with outstanding debts, those creditors have a right to seek payment from the assets of the estate. Some debts, like a mortgage, will be satisfied on the sale of the real property that secures them. Others, like credit card debts or personal loans, can only be satisfied by the assets of the estate.
The statute of limitations for a creditor to seek payment from the estate is one year after the date of death. They do, however, have a limited ability to file late if they can show good cause. They also have standing to open a probate matter if the heirs do not do so within a year of the date of death.
If your loved one dies with more debts than assets, you can have the estate declared insolvent. In that case, the creditor will have no recovery, and cannot seek payment from the heirs outside of the estate process.