If you're generating suitable income, which many college students are not, and have a suitable credit history, which many college students do not. Credit card companies want to make sure they'll be repaid. It's pretty common for them to require parental co-signers (or just issue an additional card on the parent's account) for people who don't have a full-time job.

For a debit card - yes, you can just open a bank account and get one. You need money in the bank account. If you arrange a way to automatically transfer money into a kids' bank account at the touch of the button, you end up with an app much like this postmortem.