Your response will be your personal decision, but there are some things to take into consideration before you either agree or disagree.
Tattoos are permanent. Even with removal procedures, such as lasers, being as available and effective as they are, a teenager should be impressed with the fact that tattoos should be considered a permanent work of art. This artwork will be on his or her skin for years to come, and any potential removal will be painful and expensive. A trendy haircut will grow out if your teen no longer likes it, but a tattoo is there for years, even decades.
Tattoos are very popular these days, and might be mistaken for another fad by a young person. An impressionable teen who wants to be like his or her friends, or like someone famous he or she admires, may be tempted to rush into having a tattoo without considering what it may mean in the future. The trendy symbol he wants tattooed permanently on his back today may look pretty uncool in a few years. The love of her life whose name she wants inked on her shoulder now may turn out to be the dirtbag whose name she never wants mentioned again in a few months.
If your teen is talking to you about getting a tattoo, take heart that he or she is looking for your approval and hasn't rushed out to get one illegally, or from someone completely unprofessional and unsafe, without talking to you first. Take the opportunity to urge your teen of the dangers of going to anyone other than a reputable tattoo artist. There are many risks involved with tattoos. An infection from unclean practices can ruin the tattoo and result in permanent scars. Blood-borne pathogens such as Hepatitis-C and even AIDS can be transmitted by tattoo needles and equipment when the proper precautionss aren't taken.
From there, it's your call what to do next. Your underage teen is still an underage teen, and as such, needs your approval to get a tattoo. A reputable tattoo parlor will want to you be there along with your teen, and will require you to sign a consent form in front of them before they will do the tattoo. Unfortunately, not all parlors or artists (or unscrupulous practictioners calling themselves or aspiring to be artists) will follow the legal requirements. If your teen is determined, he or she may find one of these people and get that tattoo anyway.
You may choose to go ahead and sign that consent form just to help your child avoid the dangers of a risky tattoo. Or you can urge him or her to wait until the age of consent is reached and do it later. This is where the strength of the relationship you have with your teen will be key. Good luck!