So co-signing (the having your kids on your card you mentioned) is just sharing the responsibility. Basically everyone signed on the line of credit (loan, credit card, etc) is considered equally responsible and expected to contribute financially to pay for the line of credit. Most commonly it’s a husband and wife buying a house, you both contribute financially towards the loan and you both reap the benefits of the loan, and you’re both in trouble if the loan isn’t paid, and it affects both of your scores depending on how well you manage that line of credit.

There’s different kinds of lines of credit you can get, from really bad ones like personal loans and credit cards (both are very high interest and not recommended) to moderately decent ones like car loans (interest rate is okay, but the item you’re buying loses value) to good ones like a mortgage for a house or a Home Equity Line of Credit (HELoCs are weird, but it’s using the equity on your house for what spends like a credit card but is paid back like a mortgage)