You shall not make any gashes in your flesh for the dead or tattoo any marks upon you: I am the Lord. (NRSV, Leviticus 19:28)
However, there are problems with that interpretation. Leviticus 19:25-31 contains a series of laws that prohibit various pagan worship practices. The Israelites were constantly tempted to lapse into the pagan practices of other ancient peoples ( Exodus 32:1-6, Numbers 25:1-3, etc.), so they were forbidden to do anything that had even the appearance of a pagan ritual.
Cutting and marking the flesh were common pagan rites for mourning the dead. So, in context, it was probably not the mark on the skin itself that was offensive, but rather the strong association with pagan worship. Such an association would not likely be made in today's world.
Or do you not know that your body is a temple of the Holy Spirit within you, which you have from God, and that you are not your own? For you were bought with a price; therefore glorify God in your body. (NRSV, 1 Corinthians 6:19-20)
However, in context, it is clear that 1 Corinthians 6:13-20 is about sexual immorality, not about tattoos.
Another consideration is that at least of a third of people suffer "tattoo regret" after a few months or years, and a tattoo may make a negative impression on future employers and future friends. Tattoo removal is expensive, not always completely effective, and can leave scars.