It's reasonably well known that Judaism forbids tattoos, though few people know why this law was originally instituted and why tattoos still don't make much philosophical sense today. Like many things in Judaism, the reason as stated in the Torah doesn't necessarily connect with a relevant, modern mindset, at least not directly. This doesn't mean the rule is outdated, exactly. Let's take a closer look at the prohibition of tattoos in Judaism.
The Torah is rather plain about tattooing. In a passage in Leviticus it states in as few words as possible that the skin of Jews is not to be tattooed for any reason. It doesn't explicitly explain why, though it's clear by reading the rest of the portion that tattoos are forbidden because they were associated with polytheistic cults of the ancient world. Specifically, they were associated with the most violent cults that practiced, among other things, intentional scarification and human sacrifice. These acts were then just as they are now so counter to Jewish values that any association with them is unthinkable.
Of course, idol-worshiping cults of human sacrifice haven't been part of any society surrounding Jews in a very long time. Tattoo art certainly hasn't been associated with such groups in a very long time. Granted, tattoos in Western society haven't been entirely socially acceptable for more than a generation at this point, but the continued Jewish prohibition of tattoos no longer really comes from how society in general views them.
For Jews, tattoos have come to be associated with something far more sinister than the ancient cults of sacrifice. In the Nazi concentration camps of the Holocaust, prisoners had their institutional serial numbers tattooed on their arms. It was part of the initial round of humiliations visited upon anyone unlucky enough to find themselves in the likes of Auschwitz, Dachau and Bergen-Belsen. The tattoos came after people will transported to the camps in cattle cars, their possessions taken, their heads shaved while they were stripped naked and examined by doctors as if they were livestock. The tattoo was a brand, a very real way of individuals' names being stolen in a systematic attempt to dehumanize them. For nearly a century now, a tattoo to a Jew has been a symbol of physical and spiritual genocide.
Though even if modern Jews decide that choosing a tattoo rather than having one given to them is a symbolic act of individualism and freedom, it's still not in keeping with Jewish philosophy. Even without the association of the Holocaust, tattoos are things of permanence. They are static while everything else changes. Judaism is a philosophy of growth and change, of the natural cycles of life and the humility of accepting that nothing in this world lasts forever. To get a tattoo is to tacitly claim that what one wants today is what one will always want, that what one believes today is what one will always believe. In a sense, it is a vow, a promise, things a Jew is not supposed to make because to be human is to be fallible and so there is no vow a person can keep with complete certainty.