Lev. 19:26-30, "You shall not eat anything with the blood, nor practice divination or soothsaying. 27 You shall not round off the side-growth of your heads, nor harm the edges of your beard. 28 You shall not make any cuts in your body for the dead, nor make any tattoo marks on yourselves: I am the Lord. 29 ‘Do not profane your daughter by making her a harlot, so that the land may not fall to harlotry, and the land become full of lewdness. 30 You shall keep My sabbaths and revere My sanctuary; I am the Lord."
As you can see, tattooing is mentioned in the context of ancient pagan practices. But does this mean that tattooing is in itself a bad thing to do because unbelievers did it? We have to be careful not to make a logical error known as the genetic fallacy. This error says that something is false because of its origin. For example, it would be a fallacy that because the Nazi regime developed the Volkswagen Beetle, that the VW Beetle is a bad car. Likewise, just because unbelievers get tattoos does not mean Christians should not.
The Leviticus context also tells the Israelites to not round off the hair on the side of their heads nor shave the edges of their beards. Apparently, this is what the Egyptians did and God's people were called to be separate from them. But, why such detail about hair? Perhaps Moses was concerned about the superstitious beliefs of the Egyptians to which shaving the head and beard had a religious significance. If this were the case, then God's people were certainly called to be separate from the false nations and their superstitions.
But, such religious practices that require tattooing, cutting one's own skin, and/or shaving the head and beard are not around in present-day Western culture. So, the admonition against tattooing, based upon the cultural norm, can't apply for today.
If we must conclude that tattooing is automatically bad in and of itself, then we must also conclude that shaving off the hair on the side of the head and shaving edges of men's beards is also equally bad. But, since cutting hair and shaving beards is not sinful in itself, how could we conclude that tattooing is also automatically sinful? We can't.