Answer: First, let us cover a brief definition and some background before delving into what the Bible says about tattoos. They are a decoration, pattern or mark made on the skin by puncturing or cutting it and then filling these places with a coloring substance (e.g. ink). The word tattoo is Polynesian in origin and came into existence around the 1760 - 70s (dictionary.com).
Wearing tattoos has become popular and fashionable over the last several years. Their use by individuals has expanded far beyond those in the military or in the entertainment fields.
The acceptance and expansion of "skin art" has no doubt been aided by the growth of establishments offering such embellishments. According to one advocacy site, 21,000 "tattoo parlors" currently exist in the United States alone.
The King James Bible, written in 1611, specifically describes a tattoo only in one place, Leviticus 19:28. It does not mention it by name in either the Old or New Testaments. Many modern translations, however, such as the NKJV, NIV, HCSB and the HBFV, do mention it by name.
You shall not make any cuttings in your flesh for the dead, nor tattoo (Hebrew: nathan, Strong's Concordance #H5414, meaning to add or apply) any marks (Hebrew: kethobeth, Strong's #H3793, meaning a letter or mark branded on the skin) on you. I am the Lord (Leviticus 19:28, HBFV).
God gave the above warning to the Israelites before they entered Canaan and the land they were promised as an inheritance. He did this because they were going into a land filled with people who did not worship him and who used a variety of methods (idols, markings on their body, etc.) to worship or honor their pagan deities. The Eternal did not want his people polluting themselves either by pursuing these false deities or by adopting pagan customs and using them to worship him (see Exodus 32).
The question that now arises is whether Leviticus 19:28 prohibits all kinds of tattoos or a certain subset of them. Admittedly, trying to determine whether the Bible completely condemns certain acts or merely their misuse can be difficult.
While certain acts, like incest, are clearly unacceptable under all circumstances, others, like eating or drinking alcohol, are permitted and only condemned when they are misused (e.g. gluttony and drunkenness).
The Bible clearly states that God's will is for all humans to worship him alone (Exodus 20:3, Deuteronomy 8:19, etc.). The worship of anyone or anything else, through tattoos or other means, is prohibited and a sin.
It is also clearly unacceptable to adopt the practices and symbols used to worship false gods and adapt them toward worshipping the Eternal (e.g. holidays such as Christmas, the use of crosses and so on, Deuteronomy 12:30 - 31). Idolatry, which seeks to redirect, pervert or dilute reverence due solely to God, is also wrong (Exodus 20:4 - 5).
At a minimum, based on the above principles, tattoos that depict religiously based themes or symbols (crosses, devils, pictures of Jesus or Buddha, etc.) are unacceptable.
What is unclear is the use of non-religious imagery in tattoos. The Bible does allow people to wear jewelry (rings, earrings, chains, etc.) or women to wear makeup, in order to enhance their appearance. Christians, in the final analysis, can never go wrong by prayerfully asking God for his will regarding any tattoo they may wish to place on their bodies.