'You shall not make any cuts in your body for the dead nor make any tattoo marks on yourselves: I am the LORD. (Leviticus 19:28)
The literal commandment is that we should not make any tattoo marks on ourselves.
Messiah implicitly affirmed this commandment when He spoke about the Law:
"Do not think that I came to abolish the Law or the Prophets; I did not come to abolish but to fulfill. For truly I say to you, until heaven and earth pass away, not the smallest letter or stroke shall pass from the Law until all is accomplished. Whoever then annuls one of the least of these commandments, and teaches others to do the same, shall be called least in the kingdom of heaven; but whoever keeps and teaches them, he shall be called great in the kingdom of heaven. (Matthew 5:17-19)
This commandment indicates that we should not depart from the way G-d has ordered His creation. Our flesh is formed exactly the way that G-d intended for it to be. This commandment pictures Messiah's complete obedience to G-d's order for creation.
Even in His greatest grief (Matthew 26:38) He sought the Father's will:
And He went a little beyond them, and fell on His face and prayed, saying, "My Father, if it is possible, let this cup pass from Me; yet not as I will, but as You will." (Matthew 26:39)
Messiah knew His role in Creation and stood firm even in the face of execution to remain in that role and perform His work.
Messiah fulfilled this commandment by not tattooing His skin.
This commandment is observed in a very literal fashion: tattoos are forbidden. Tattoos are generally associated with pagan practices and idol worship and were thus forbidden.
We are able to fulfill this commandment today and we should not tattoo our skin.
Liberals from both Christianity and Judaism claim that the commandment prohibiting tattoos only applies to getting tattoos "for the dead" as is noted in the first sentence of the verse. The grammar of the Hebrew text associates the "for the dead" phrase with the prohibition against cutting the flesh. It does not apply to "putting on yourself any etched mark". Their claim is not supported by text of the passage.
Although we live in a fallen, sinful, and broken world, our bodies are made exactly the way G-d wants them to be... for now. Imagine taking a priceless painting by the master artist Picasso or Rembrandt and doodling an addition to it. Consider taking a statue by Michelangelo, Rodin, or other famous sculptor and sketching a picture on it. Ponder the idea of taking a musical masterpiece by Beethoven, Tchaikovsky, Hayden, Handel, or Chopin and just adding a few random notes here and there.
When a person tattoos their body they suppose they have a better idea of how their body should look than the Master who created it.