The source of this prohibition is Leviticus 19:28: "You shall not etch a tattoo on yourselves." This prohibition applies to all tattoos besides those made for medical purposes such as to guide a surgeon making an incision.
Although some of the commentaries1 seems to believe that this is one of the Torah's chukim--the commandments whose rationales transcend the ken of human intellect--other commentators do offer several explanations for this prohibition:
1) The human body is G‑d's creation, and it is therefore unbefitting to mutilate G‑d's handiwork. It is especially unbefitting for members of G‑d's chosen nation to mutilate their bodies. One must believe that G‑d, the greatest artisan of all, formed him or her in the most fitting way, and one must not change this form. Changing one's body (unless it is for the health reasons) is tantamount to insulting G‑d's handiwork.2
2) In ancient times it was customary for idol worshippers to tattoo themselves as a sign of commitment to their deity—much like an animal that is branded by its owner. On many occasions the Torah forbids practices that emulate pagan customs—considering that following their traditions is the first step towards ascribing to their idolatrous beliefs and services.3
3) The covenant of circumcision is unique in its being a sign in our bodies of our relationship with G‑d. Making other signs in one's body would weaken and cheapen this special sign.4
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