Under the Law of Moses, the Israelites were commanded
You shall not make any cuts on your body for the dead or tattoo yourselves: I am the Lord. (Leviticus 19:28)
Although the New Testament does not prohibit tattoos, many Christians object to tattoos because of this verse. After all, if God banned tattoos then, we should at least consider whether there it is right for a believer to have a tattoo now.
First, we need to understand why the law was given. Cutting oneself appears to have been a pagan practice to attract a god’s attention. For example, the prophets of Baal cut themselves when Elijah challenged them to demonstrate that Baal was real (1 Kings 18:28). To cut yourself “for the dead” was probably a way of warding off the spirits of the dead or of demonstrating your deep grief to your god. Similarly, tattoos were probably associated with pagan idolatry, and may have been used to protect a person from the spirits of the dead. Some evidence for this has been found in human remains in Scythia dating to the six century BC.
The Israelites were consistently told to avoid pagan worship, and to avoid practices that were associated with pagan worship (e.g., Leviticus 21:5). These days, tattoos are not associated with pagan worship, and so the reason behind the law is no longer relevant.
On the other hand, Paul tells us that we should treat our bodies as belonging to God.
1 Corinthians 6:19-20
Or do you not know that your body is a temple of the Holy Spirit within you, whom you have from God? You are not your own, for you were bought with a price. So glorify God in your body.
Permanently marking our bodies is not the best way to look after what God has given us, and so tattoos are probably best avoided.