You shall not make any cuts on your body for the dead or tattoo yourselves: I am the LORD. (ESV)
But this prohibition was centered around forbidding the Israelites to mimic the worship practices of their pagan neighbors. Since modern tattoos are not connected with pagan worship practices it really doesn’t transfer.
I think the best consideration for either getting or not getting a tattoo is 1 Corinthians 6:19-20 which says:
Do you not know that your body is a temple of the Holy Spirit, who is in you, whom you have received from God? You are not your own; you were bought at a price. Therefore honor God with your body.
The first consideration every Christian ought to have concerning a tattoo is whether the Lord has given the go ahead. According to the passage above, your body no longer belongs to you. It’s His—bought and paid for by His blood—so you need to make sure He’s okay with it.
Secondly, any consideration for a tattoo needs to be in keeping with the exhortation to “…honor God with your body.” Is the tattoo something that will bring honor to the Lord?
Finally, Christians are to be influenced by the Word—NOT the World! If the motivating factor for getting a tattoo is simply because they’re popular then you might want to reconsider your decision. As a believer, you are to be led by the Spirit of Christ, not the spirit of the world.
Now, I understand that these comments don't touch on the issue of someone who has gotten a tattoo which they now regret. It's not uncommon for an individual to come to Christ after having received one or more tattoos of questionable or even objectionable content. What does a person do in such a case?
Looking at a tattoo that only reminds you of your foolish past is probably similar to the memory the Apostle Paul retained of his own life before Christ. It wasn't something he was proud of, and he even wrote to the Corinthians about it saying:
For I am the least of the apostles, unworthy to be called an apostle, because I persecuted the church of God. 1 Corinthians 15:9 (ESV)