Yes, you could “authorize” another person to carry and use a debit card linked to your account. Since an authorized user would be able to withdraw money from your account at any time, be sure he or she is trustworthy and responsible before you provide a card. In some cases, it might be better to help the other person set up an account in their own name rather than adding them to your account. (This would also help avoid the risk of them making transactions on your account that you're not aware of and that could potentially cause you to overdraw the account and incur overdraft or "bounced" check fees.)
Read Consumer Action's Checking and Savings Accounts publication" to learn more about opening an account.