I have read so much controversy over this topic, when really it is such a minor point in the Bible [1]. There are only a few verses that seem to relate to it, but there is a lot of discussion generated. A search on Google for "can christians get tattoos" returns about 4,930,000 results!
Leviticus 19:27-28
The Bible verse most often pointed to on this topic is Leviticus 19:27-28. In the New International Version it says "Do not cut the hair at the sides of your head or clip off the edges of your beard. Do not cut your bodies for the dead or put tattoo marks on yourselves.". However, it's important to understand that many of the rules given to the Israelites in Leviticus were designed to keep them from adopting the practices of the nearby pagan tribes. God didn't want them to be corrupted or led into worshiping other gods, so he made rules against things like tattoos and certain hairstyles because it showed that the person was a follower of another god. Now, these certain things are no longer associated with worshiping other gods, which is why it's okay for us to cut our hair anyway we want, can wear clothes made with two different fibers, and can eat rare steak - all things that were forbidden in Leviticus.
Origin and Significance of Tattoos
There is also the origin of the practice of tattooing to consider. It is often mentioned that tattooing came from pagan cultures, can be associated with occultism, and was originally popular with more undesirable aspects of society (criminals, early sailors, etc.) [2]. In 1 Thessalonians 5:22 it says "Accept what is good and don't have anything to do with evil.". Some Christians say that since tattoos were once associated with pagans, occultists, and criminals that Christians should not get them. However, it seems that this is more of a cultural phenomenon - many young people today don't have this association and tattoos aren't seen as an indicator of crime or evil in our culture.
Your Body is God's Temple
Another important verse which is often referenced when talking about tattoos is 1 Corinthians 6:12-20. Here it says "You surely know that your body is a temple where the Holy Spirit lives. The Spirit is in you and is a gift from God. You are no longer your own. God paid a great price for you. So use your body to honour God.". This can be taken in two ways - either the tattoos should be honorable to God, or you should not get tattoos. Christians who are against tattoos see it as putting graffiti on God's Temple. Just as you wouldn't go to a famous art museum and try to modify the beautiful works of art, they say you shouldn't do anything to modify your body. Theoretically then, these people should also be against piercings, hair dye, and makeup, because anything that modifies the body is changing it from how God designed it to be. However, many Christians who hold that the body is a temple probably do use hair dye, and makeup, and nobody complains about lower ear piercings for girls. So that seems to me that they are being hypocritical. Jesus often condemned the Pharisees for being hypocrites and judging others for the tiniest parts of the law while ignoring the more important aspects such as mercy and faithfulness (Matthew 23:23-24).
Don't Judge People For Their Tattoos
There are some Christians who go so far as to judge others for having tattoos. In the book "I'm Fine With God - It's Christians I Can't Stand"[3], there is a story of a Christian doctor who refused to treat a girl with an ear infection because her mother had a tattoo. Some accuse Christians with a tattoo of being insincere, sinning, or "backsliding" (falling away from Jesus). One thing is certain - Christians should not judge others based on if they have a tattoo or not. "Don't judge others, and God won't judge you. Forgive others, and God will forgive you." (Luke 6:37). Christians should be much more concerned with the state of a person's soul, not the state of the upper layer of their skin. Christians who judge others for something as minor as a tattoo, while ignoring the fact that we are told to love others are like the Pharisees Jesus talked about in Luke 6:41. Jesus says to them "You can see the speck in your friends eye. But you don't notice the log in your own eye." as a way of saying that you should be sure that you should examine your own heart and intentions before you try to correct others for a much smaller mistake.
Don't Be Like the World
Others use Mark 9:50 as a justification for not getting tattoos, or anything else that seems to blend in with society. "Salt is good, but if it no longer tastes like salt, then how can it be made salty again?". They say that Christians should be in the world, but not of the world (John 17:14-17). And that if we blend into society (by tattoos, fashion, acting the same as others, etc.) then that we will not stand out and make others curious about why we are different. However, some younger Christians like the idea of getting a tattoo with some religious symbolism as a way of showing that they are owned by God. Others use them as interesting conversational pieces that perhaps will let them talk to others more easily about their faith [4],[5].
How to Tell if Tattooing is OK
I personally believe that a tattoo is not evil, and is not necessarily wrong. It depends more on the person's heart, and what they believe. If they believe it is wrong to get a tattoo, then they shouldn't because it would be going against what they think God has told them. This relates to what Paul said in Romans 14:14 "The Lord Jesus has made it clear to me that God considers all foods fit to eat. But if you think some foods are unfit to eat, then for you they are not fit.". Further details are given in the entire chapter of Romans 14. Here it explains how Christians can determine what is right for minor issues that are not specifically discussed in the Bible. It doesn't apply to the 10 Commandments, or any of the rules that Jesus specifically stated. But it can relate more to issues that Christians face today - such as types of music, movies, dancing, drinking alcohol, etc. which are not specifically talked about in the Bible. The advice given by Paul is that if you feel it's okay and you have prayed about it and it doesn't make you sin or turn away from God, then it's okay for you. But if you do it in the presence of someone who is affected by it, then you should avoid it around them to not make them feel like they are sinning. For example, don't drink alcohol around a recovering alcoholic. So for tattoos, if you know that someone who looks up to you or who is close to you is seriously affected negatively by tattoos for whatever reason, maybe it's a better idea not to get one.
Concluding Thoughts
So if you are okay with getting a tattoo - you have prayed about it, you have considered it carefully, it is not of a subject dishonourable to God, and you accept that others may judge you for it, then it's probably okay for you to get one. But if it's just to show off, or to be rebellious, then it's probably not a good idea. It's important to note that tattoos are permanent, and will be with you the rest of your life (or will cost a lot and will be painful to remove). They will also stretch and deteriorate over time, and you may regret them later on depending on how you or your life situation changes. There is also the chance of getting diseases from the method of tattooing. So please consider carefully your decision.
Another good article about tattoos and what to consider as a Christian is here [6].
[1] http://christiannews.christianet.com/1199208432.htm
[2] http://www.raptureready.com/featured/schaefer/tattoo.html
[3] "I'm Fine With God - It's Christians I Can't Stand" by Bruce Bickel and Stan Jantz, p.35
[4] http://www.revelife.com/711740604/christian-tattoos-my-tattoo-tour-with-pics/
[5] http://www.gotquestions.org/Christian-tattoos.html
[6] http://christianity.about.com/od/faqhelpdesk/f/tattoochristian.htm