Great question. Financial counselor and talk radio host Dave Ramsey often says "What you own stands good for what you owe.”
This means that the estate (all that the person owned - their assets, their property, etc) will be used to pay off the outstanding debt before it is divided up amongst other heirs.
For instance, if a husband has a balance on his personal credit card (and his wife was not a co-signer on it), and the husband passes away, the surviving wife does not take on his debt. Rather, the assets that are considered his will be used to pay off the debt, such as up to 50% of any cash in the bank and 50% of the sale value of the house.
But it's important to understand that debt is not transferable. You can't inherit a debt from your deceased spouse or parent. And a deceased relative's credit standing will not affect your own credit standing. Debt collectors may try to convince you otherwise, but you're not liable for a debt you did not incur. The only way you can incur debt is if you signed for it, be that through your personal credit account or an account you co-signed for.
Certainly in instances where the husband and wife were co-signers on the same credit card, or where the husband and wife share ownership of a house together, things can get complicated. It's always a good idea to talk with a competent estate attorney who can sort through the issues with you.
Bottom line, don't pay for anything you're not convinced you're responsible for. Collectors will try to collect from anyone, so stand firm until you know what the absolute right answer is.