Credit card debt does not pass to your family when you die. It is paid by your estate, the money and property you leave behind. If the estate cannot cover the balance, the rest usually goes unpaid, and most relatives owe nothing.[1]
That is the answer for the large majority of people. There are exceptions, and they are the reason this question causes so much fear: a joint account holder, a co-signer, and in some states a surviving spouse can owe the balance. An authorized user cannot. This page spells out who pays, who does not, what happens when the money runs out, and what a debt collector is and is not allowed to say to you.