Good question, Rob. I'll give you the nutshell answer but we can talk more if you like..
First, I’ll note a very important distinction in the Old Testament. We refer to “the law of God” and “the law of Moses.” The law of God is for all peoples at all times. The law of Moses, however, was given directly to the Israelites for that time in history. Chapter 19 of Leviticus begins by directing the content to the Israelites. Notice that just a couple sentences before the text that you mention. It forbids from shaving around the sides of your head and also forbids from wearing clothes mixed with linen and wool. Interesting. Today’s Christians are not held under these laws because they were not given to us. We study about them, learn from them, and apply their spiritual principles to our lives.
Next, it’s important to understand the context of what was going with the Israelites when this law was given. The Israelites weren’t getting butterflies and tribal designs tattooed. They were actually tattooing idols onto their bodies as a means of idol worship. They were also cutting scars into their bodies in an attempt to stay connected with the dead. This is why God forbade this practice. Tattooing in our modern culture spans a broad range of intentions and purposes. If a Christian tattooed his body with a Buddha, I’d be a bit concerned. However, a Christian tattooing his body with a beautiful peace of artwork that represents a spiritual truth in his life is different. As Christians, we are no longer under the laws of the Old Testament. Rather, we are under “the law of love” and we have freedom in Christ to enjoy that.
At the end of the day, I would never argue with a guy that believed tattooing was Biblically wrong. I’d just tell him “fine, don’t get a tattoo.” However, I would argue that the Christian that doesn’t have an issue in his conscience with getting a tattoo has the freedom in Christ to do so. Many verses in the Bible back that up and I’d be glad to expound if you’re interested. (Guess that was a big nutshell!). Hope that helps.