Tattoos are serious business. They're painful, expensive and permanent (or even more expensive to get removed later on). If you don't believe getting a tattoo is appropriate for your teen, this is a family standard you should make aware to him/her early on. While your teens might think getting a tattoo is a great way to assert their independence and create a new identity, they can't get one without your permission until they reach age 18.
Even if you set the expectation in your teen that getting a tattoo isn't acceptable, it doesn't mean he/she isn't going to beg and bother you about it. Do give your teen the respect of discussing the issue with him/her so he/she understands why you're saying no on this issue. Discuss medical risks, such as Hepatitis, HIV, infections and scarring. Your teen may also have some specific health risks if he or she has any heart problems.
You're most likely against tattoos because of their permanency, so sit down with your teen and ask him a few questions. These questions shouldn't be asked in an aggressive or judgmental manner. What are his goals in getting a tattoo? Is it because someone else has one (e.g. a celebrity or new friend)? Where on his body does he want the tattoo? What if your teen doesn't like it once it's done? How does he think it will change how people see him? How does he think having the tattoo might affect him in the future? This is a great opportunity to have a good talk with your teens about their self-image and self-esteem. If the urge passes, your teen just might be grateful you stood your ground on this issue.