If you're asking because a loved one of yours passed away, or is extremely ill, I'm very sorry to hear that, and doubly sorry if your loved one's credit card debts are causing you undue stress. But the good news is that most likely your cardholder's estate is liable for the balance and not you, or anyone else. Why "most likely?" More on that in a moment.
If a probate court is involved, they'll decide what bills should be paid and how fast, and who gets paid first. The money remaining is then generally disbursed to anyone who stands to inherit what's left.
But things get complicated if a loved one dies and there is a co-signer on the account. Then, even if it was always understood that the deceased was making the payments, the co-signer is equally responsible for paying off the credit card debt. Now, if you were simply an authorized user -- you never signed the application for the card and you were placed on there as someone who was allowed to use the card -- you're in the clear.
Bottom line, if you didn't sign the credit card application, you aren't responsible for paying it off. And, remember, while there are a lot of principled debt collectors out there, there are plenty of unscrupulous ones who will appeal to your emotions and talk about how sad it would be for your father or aunt or whomever to leave a legacy of unpaid bills. But before you let that argument get to you, remember: the debt collector simply wants the money, one way or another, and it seems extremely doubtful that anyone on their deathbed is thinking, "Boy, when I expire, I hope my kid or one of my nephews can go broke paying off my credit card to a debt collection agency." That's the legacy that I'm sure nobody wants to leave behind.