When a family member dies, his estate is responsible for paying his remaining taxes, outstanding loans and credit-card balances. If the deceased didn't leave enough money and assets to pay his bills, the card company may insist you pay them instead. In most cases they have no legal right to get a penny from you.
The deceased's executor must notify creditors about the death, using whatever methods state law requires. If the credit-card company doesn't reply by the state's deadline -- usually four to six months -- it loses its right to get paid. Even if the reply is timely, the estate may have to pay other creditors first -- taxes come ahead of credit-card bills, for instance -- and the company may therefore see only partial payment, or no money at all.
The card company has a stronger hand if you and the deceased had a joint account, or if it's your account and he was an authorized user. If you made yourself legally responsible for his debts, the company should still look to the estate first. If that well runs dry, though, they can come after you, even if the bill is entirely the deceased's spending. If you were an authorized user on the deceased's account, you're not liable for his bills, but the company may demand you pay up anyway.
If your spouse dies and you live in a community property state, that adds another wrinkle to the law. Community-property laws treat the bills one spouse runs up during the marriage as joint debts: you're responsible for his debts and vice-versa, so his creditors can sue you for whatever the estate doesn't pay. You aren't liable if he rung up the debt before the marriage, or if you have a written agreement to keep your debts separate.
Creditors know you're vulnerable when you're mourning. They may call, hoping that guilt, pride or weakness will convince you to pay off the card. If you aren't legally obligated, however, they can't do anything to make you pay. When the caller is a collection agency rather than the card company itself, federal law gives you the upper hand. You can tell the collector not to contact you again and report her to the Federal Trade Commission if she does. You can also report her if she lies about your legal obligation for the debt.