In the context of your credit card, an authorized user is someone to whom you've granted financial access. When you add an authorized user to your account, they get their own credit card with their name on it. However, only the primary cardholder (you) will receive the bill.
Depending on your card issuer, you may have access to account settings that let you limit the amount of money your authorized user can spend each day or each month. Generally speaking, you should also be able to set other stipulations on your authorized user's account, such as limiting or denying cash advances made with the card.
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By adding your teen (or anyone else) as an authorized user, you are taking full responsibility for any purchases they make. In other words, it's up to you to set the ground rules to make sure everything goes as planned.
While it's easy to assume that teens and credit cards don't mix, adding your teen as an authorized user on your account comes with four benefits: